problem solving volume of a cube

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Volume of a cube

Here you will learn about the volume of a cube, including how to calculate the volume of a cube within mathematical problems and within real-world contexts.

Students will first learn about volume of a cube as part of measurement and data in 5th grade and extend their learning as part of geometry in 6th grade.

What is the volume of a cube?

The volume of a cube is the amount of space there is within a cube.

A cube is a three-dimensional shape with 6 square faces.

To find the volume of a cube, with side length a, you can use the volume of a cube formula, \text {Volume }=a^{3}.

Volume is measured in cubic units. For example, cubic inches (in^3), cubic meters (m^3), or cubic centimeters (cm^3).

For example,

The volume of this cube is,

volume = a^3

volume = 8^3

volume = 512 \, cm^3

The length, width, and height of the cube are multiplied together to find the total volume.

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5 – Measurement and Data (5.MD.3) Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
• Grade 5 – Measurement and Data (5.MD.4) Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
• Grade 5 – Measurement and Data (5.MD.5) Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, for example, to represent the associative property of multiplication. b. Apply the formulas V = l \times w \times h and V = b \times h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
• Grade 6 – Geometry (6.G.2) Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l \times w \times h and V = b \times h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

How to calculate the volume of a cube

In order to calculate the volume of a cube:

Substitute the values into the formula.

Work out the calculation.

Write the answer and include the units.

[FREE] Volume Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!

Volume of a cube examples

Example 1: volume of a cube.

Find the volume of the cube.

Write down the formula.

\text{Volume }=a^{3}

2 Substitute the values into the formula.

Here, the sides of the cube are 6 \, cm.

\text{Volume }=6^{3}

3 Work out the calculation.

\begin{aligned} \text{Volume} &=6 \times 6 \times 6\\\\ &=216 \end{aligned}

4 Write the answer and include the units.

The measurements are in centimeters. Therefore, the volume will be in cubic centimeters.

\text{Volume }=216 \mathrm{~cm}^{3}

Example 2: volume of a cube

Find the volume of this cube.

Here, the edges are each 7 \, in.

\text{Volume }=7^{3}

\begin{aligned} \text{Volume} &=7 \times 7 \times 7\\\\ &=343 \end{aligned}

The measurements are in inches. Therefore, the volume will be in cubic inches.

\text{Volume }=343 \mathrm{~in}^{3}

Example 3: volume of a cube – different units

Notice here that one of the units is in centimeters while the other is in meters. You need all the units to be the same to calculate the volume.

This is a cube, so you know all the edges are the same length, so you can easily change meters to centimeters.

0.6 \mathrm{~m}=60 \mathrm{~cm}

\text{Volume }=60^{3}

\begin{aligned} Volume &=60×60×60 \\\\ &=216,000 \end{aligned}

\text{Volume }=216,000 \mathrm{~cm}^{3}

Example 4: volume of a cube – different units

Notice here that one of the units is in meters, one is centimeters, and another is in millimeters. You need all the units to be the same to calculate the volume.

This is a cube, so you know all the edges are the same length, so you can easily change the centimeters and the millimeters to meters.

\begin{aligned} & 15,000 \mathrm{~mm}=15 \mathrm{~m} \\\\ & 1,500 \mathrm{~cm}=15 \mathrm{~m} \end{aligned}

\begin{aligned} \text { Volume } & =15 \times 15 \times 15 \\\\ & =3,375 \end{aligned}

The measurements are in meters. Therefore, the volume will be in cubic meters.

\text{ Volume }=3,375 \mathrm{~m}^3

Example 5: volume of a cube – word problem

Anna has a Rubik’s cube. Each edge of the Rubik’s cube is 5.8 \, cm. What is the volume of the Rubik’s cube?

Here, the edges are each 5.8 \, cm.

\text{ Volume }=5.8^3

\begin{aligned} \text { Volume } & =5.8 \times 5.8 \times 5.8 \\\\ & =195.112 \end{aligned}

\text { Volume }=195.112 \mathrm{~cm}^3

Example 6: volume of a cube – word problem

Grant is moving to a new house and needs to buy boxes to pack his belongings into. He wants to know the volume of the cube-shaped box shown below. The length of the side of the box is 20 inches. Help him determine the box’s volume.

Here, the edges are each 20 \, in.

\text{Volume }=20^{3}

\begin{aligned} \text { Volume } & =20 \times 20 \times 20 \\\\ & =8,000 \end{aligned}

\text{Volume }=8,000 \mathrm{~in}^3

Teaching tips for volume of a cube

• Be sure to give students a wide variety of practice problems within their worksheets that incorporate cubes and other three-dimensional objects found in the real world. This will give students a deeper understanding of the concept of volume.
• Students should fully understand why the volume formula works with a cube – they need to understand why multiplying 3 of the cubes edges (length, width and height) gives us the cube’s volume. Once they have a strong understanding of the formula, they can easily use it to find the volume of any cube.

Easy mistakes to make

• Forgetting to include the units in your answer or writing the incorrect units. You should always include units in your answer. Volume is measured in cubic units. (For example, mm^3, cm^3, m^3, etc).
• Not converting all measurements to the same unit You need to make sure all measurements are in the same units before calculating volume. For example, you can’t have some measurements in centimeters and some in meters.
• Confusing volume of a cube with surface area of a cube Volume is the space inside a cube and it is a three-dimensional measurement. Surface area is the total area of each of the cube’s square faces and it is a two-dimensional measurement.

Related volume lessons

• Volume of a cylinder
• Volume of a hemisphere
• Volume of a sphere
• Volume formula
• Volume of a prism
• Volume of a cone
• Volume of a triangular prism
• Volume of a rectangular prism
• Volume of square pyramid
• Volume of a pyramid

Practice volume of a cube questions

1. Find the volume of the cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume } &=3^{3} \\\\ &=3 \times 3 \times 3 \\\\ &=27 \mathrm{~cm}^{3} \end{aligned}

2. Find the volume of the cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume }&=0.5^{3}\\\\ &=0.5 \times 0.5 \times 0.5\\\\ &=0.125 \mathrm{~m}^{3} \end{aligned}

3. Find the volume of this cube.

This is a cube, which means all the edges are the same length, so you can easily change meters to centimeters, 0.4m = 40 \, cm.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text{Volume }&=40^{3}\\\\ &=40 \times 40 \times 40\\\\ &=64,000 \end{aligned}

\text{Volume }=64,000 \mathrm{~cm}^{3}

4. Kara has 5 sugar cubes to put into her coffee. Each sugar cube has side lengths of 13 \, mm. What is the total volume of all 5 sugar cubes?

First, you need to find the volume of one sugar cube.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =13^3 \\\\ &= 13 \times 13 \times 13 \\\\ &= 2,197 \mathrm{~mm}^3 \end{aligned}

Since each sugar cube has the same side lengths, you can multiply the volume of one sugar cube by 5 to find the total volume of all 5 sugar cubes.

2,197 \mathrm{~mm}^3 \times 5=10,985 \mathrm{~mm}^3

5. Vera has two boxes, box A and box B, which are shown below. How much greater is the volume of box B than the volume of box A?

First, you need to find the volume of box A and the volume of box B. Then you need to find the difference between the two.

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =12^3 \\\\ & = 12 \times 12 \times 12 \\\\ & = 1,728 \mathrm{~in}^3 \end{aligned}

\begin{aligned} \text{Volume } &=a^{3} \\\\ \text { Volume } & =16^3 \\\\ &= 16 \times 16 \times 16 \\\\ &= 4,096 \mathrm{~in}^3 \end{aligned}

4,096-1,728=2,368

The volume of box B is 2,368 \mathrm{~in}^3 greater than the volume of box A.

6. This sculpture is formed by placing one cube on top of another. Find the total volume of the sculpture.

To find the total volume of the sculpture, you need to find the volume of the top cube and the volume of the bottom cube, then add the volumes together.

Volume of bottom cube: 60 \times 60 \times 60=216,000 \mathrm{~cm}^3

Volume of top cube: 35 \times 35 \times 35=42,875 \mathrm{~cm}^3

Total volume: 216,000+42,875=258,875 \mathrm{~cm}^3

Volume of a cube FAQs

A cube is a three-dimensional shape with 6 square faces. The edges of the cube are of equal length.

To find the volume of a cube, with side length a, you can use the volume of a cube formula, \text { Volume }=a^3. This is the same as multiplying length times width times height.

The volume of a cube formula is \text { Volume }=a^3.

The next lessons are

• Surface area
• Pythagorean theorem
• Trigonometry

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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

• Surface Area

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Last modified on August 3rd, 2023

#ezw_tco-2 .ez-toc-title{ font-size: 120%; ; ; } #ezw_tco-2 .ez-toc-widget-container ul.ez-toc-list li.active{ background-color: #ededed; } chapter outline

Volume of a cube.

The volume of a cube is the space it takes up in the three-dimensional plane. It is measured in cubic units such as m 3 , cm 3 , mm 3 , ft 3 , or in 3 . The volume of a cube determines how big it is.

With Edge Length

The basic or common formula to determine the volume of a cube is:

Let us solve some examples involving the above formula.

Find volume of a 5 x 5 cube.

As we know, The edge length determines the size of a cube, Volume ( V ) = a 3 , here a = 5 units = 5 3 = 125 cubic units

What is the volume of a 9 inch cube?

As we know, Volume ( V ) = a 3 , here a = 9 in = 9 3 = 729 in 3

Calculate the volume of a 40ft high cube container.

As we know, The length, breadth, and height of a cube is same, Volume ( V ) = a 3 , here a = 40 ft ∴ V = 40 3 = 64000 ft 3

With Diagonal

The formula to get the volume of a cube when the diagonal is known is:

Here we will use the equation of the diagonal to derive the formula of volume of a cube with diagonal.

Length of Space Diagonal (d) = ${\sqrt{3}a}$ — (1)

The equation for volume of a cube with edge length is V = a 3 , — (2)

Now, replacing the value of a in eqn (1)

a = d/√3, from (1)

V = a 3 = (d/√3) 3 replacing the value of a from (1)

= d 3 /(√3 × √3 × √3)

∴ V = ${\dfrac{\sqrt{3}}{9}d^{3}}$

(V = √3d 3 /9)

Calculate the volume of a cube given its diagonal 3 cm.

As we know, Volume ( V ) = ${\dfrac{\sqrt{3}}{9}d^{3}}$, here d = 3 cm ∴ V = √3 × 3 3 /9 = 5.2 cm 3

finding the EDGE of a cube when the VOLUME is known

Calculate the side length of a cube given its volume 343 cubic millimeters.

Side length of a cube is actually its edge length. Oftentimes the edge is termed as side length. So, we will use an alternative formula to find the edge with the volume, ${a=\sqrt[3] {V}}$, here V = 343 mm 3 ${a=\sqrt[3] {343}}$ = 7 mm

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Volume of a Cube Worksheets

Walk through this unit of printable volume of a cube worksheets hand-picked for 5th grade, 6th grade, and 7th grade students, comprising pdfs with problems presented as solid shapes and as word problems. A cube is a unique 3-dimensional shape that has squares for all six of its sides. Know the formula for the volume of the cube and use it to solve mathematical problems. Also, learn to find the side length of the cubes. Begin your learning with our free worksheets.

Volume of Cubes | Integers - Easy

Multiply the length of the given side thrice to calculate the volume. Gain a conceptual understanding of volume and solve problems presented as 3D shapes and in word problems with dimensions involving integers ≤ 20.

• Download the set

Volume of Cubes | Integers - Moderate

Add-on to your practice and level up with this batch of printable volume of a cube worksheets for grade 5. Find the volume of each cube whose side lengths are presented as 2-digit integers.

Volume of Cubes | Decimals

Plug in the measure of the side length (a) in the volume of a cube formula V = a 3 to determine the volume of the cube. The side length is expressed as decimals. Compute and round off the answer to two decimal places.

Volume of Cubes | Fractions

Convert mixed fractions to improper fractions if required and then multiply side length thrice presented as fractions to figure out the volume enclosed by each cube in these pdf worksheets for grade 6 and grade 7.

Finding the Side length of the Cube

Solve for the side length or the edge of the cube by rearranging the volume of the cube formula. Determine the cube root of the given volume to find the length of the side. Round off to the nearest tenth.

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» Volume by Counting Cubes

» Volume of Rectangular Prisms

» Volume of Triangular Prisms

» Volume of Prisms

» Volume of Mixed Shapes

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Volume of a Cube

Learn how to find the volume of a cube., volume of a cube lesson, cube volume formula.

The formula for volume of a cube is given as:

Where V is the volume and a is the edge length.

Volume of a Cube Example Problems

Let's go through a couple of example problems together to practice finding the volume of a cube.

Example Problem 1

Find the volume of a cube with an edge length of 5.

• We can simply plug the edge length into the formula. This gives us:
• V = (5) 3 = 125
• The volume of the cube is 125.

Example Problem 2

The volume of a cube is 343. What is the edge length?

• Let's plug the volume into the formula and then solve for the edge length a .
• The edge length is 7.

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• Volume of Cubes – Explanation & Examples

Volume of Cubes – Explanation & Examples

A cube is 3- dimensional shape with 6 equal sides, 6 faces, and 6 vertices in geometry. Each face of a cube is a square. In 3 – dimension, the cube’s sides are; the length, width, and height.

In the above illustration, sides of a cube are all equal i.e. Length = Width = Height = a

Cubes are everywhere! Common examples of cubes in the real world include square ice cubes, dice, sugar cubes, casserole, solid square tables, milk crates, etc.

The volume of a solid cube is the amount of space occupied by the solid cube . The volume is the difference in space occupied by the cube and the amount of space inside the cube for a hollow cube.

How to Find the Volume of a Cube?

To find the volume of a cube, here are the steps:

• Identify the length of the side or length of the edge.
• Multiply the length by itself three times.
• Write the result accompanied by the units of volume.

Volume is measured in cubic units, i.e., cubic meters (m 3 ), cubic centimeters (cm 3 ), etc. We can also measure the volume in liters or milliliters. In such cases, the volume is known as capacity.

Volume of a Cube Formula

The volume of cube formula is given by;

Volume of cube =length * width * height

V = a * a * a

= a 3 cubic units

Where V= volume

a = The length of the edges.

Let’s try the formula with a few example problems.

What is the volume of a cube whose sides are 10 cm each?

Given, the side length = 10 cm.

By the volume of a cube formula,

Substitute a = 10 in the formula.

= (10 x 10 x 10) cm 3

= 1000 cm 3

Therefore, the volume of the cube is 1000 cm 3 .

The volume of a cube is 729 m 3 . Find the side lengths of the cube.

Given, volume, V = 729 m 3 .

To get the side lengths of the cube, we find the cube root of the volume.

3 √ 729 = 3 √ a 3

So, the length of cube is 9 m.

The edge of a Rubik’s cube is 0.06 m. Find the volume of the Rubik’s cube?

Volume = a 3

= (0.06 x 0.06 x 0.06) m 3

= 0.000216 m 3

= 2.16 x 10 – 4 m 3

A cubical box of external dimensions 100 mm by 100 mm by 100 mm is open at the top. Suppose the wooden box is made of 4 mm wood thick. Find the volume of the cube.

In this case, subtract the thickness of the wooden box to get the dimensions of the cube.

Given, the cube is open at the top, so we have

Length = 100 – 4 x 2

Width = 100 – (4 x 2)

Height = (100 – 4) mm…………. (a cube is open at the top)

Now calculate the volume.

V= (92 x 92 x 96) mm 3

= 812544 mm 3

= 8.12544 x 10 5 mm 3

Cubical bricks of length 5 cm are stacked such that the height, width, and length of the stack is 20 cm each. Find the number of bricks in the stack.

To get the number of bricks in the stack, divide the stack’s volume by the brick volume.

Volume of the stack = 20 x 20 x 20

= 8000 cm 3

Volume of the brick = 5 x 5 x 5

Number of bricks = 8000 cm 3 /125 cm 3

= 64 bricks.

How many cubical boxes of dimensions 3 cm x 3 cm x 3 cm can be packed in a large cubical case of length 15 cm.

To find the number of boxes that can be packed in the case, divide the case’s volume by the volume of the box.

Volume of each box = (3 x 3 x 3) cm 3

Volume of the cubical case = (15 x 15 x 15) cm 3

= 3375 cm 3

Number of boxes = 3375 cm 3 /27 cm 3 .

= 125 boxes.

Find the volume of a metallic cube whose length is 50 mm.

Volume of a cube = a 3

= (50 x 50 x 50) mm 3

= 125,000 mm 3

= 1.25 x 10 5 mm 3

The volume of a cubical solid disk 0.5 in 3 . Find the dimensions of the disk?

a = 0.794 in.

Practice Questions

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Volume of a Cube

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Volume of a Cube Calculator

What is a cube, what is the volume of a cube, how to calculate the volume of a cube (by hand).

• How to use Omni's volume of a cube calculator?

Omni calculator advanced options: find the volume of a cube without knowing the side.

Why is the formula for the volume of a cube so simple, cubes in the world from ice cube to icecube and beyond.

Welcome to Omni's volume of a cube calculator . Ever wondered what is the volume of a cube or how come the formula for the volume of a cube is so simple? Well, we did, and we've got the answers. Here we explain how to calculate the volume of a cube while also taking a look at what makes the cube such a popular shape.

🔎 With our volume calculator , you can find the volume of many other 3D shapes.

Let's start from the beginning. A cube is a 3D object made up of 6 faces, all of which are squares of equal size . If you want to go down this particular rabbit hole, we can say that squares are also regular objects, this time in 2D space, made up of 4 segments of equal length meeting at 90-degree angles.

A cube is one of the most basic 3-dimensional objects, together with the tetrahedron (a regular triangular pyramid) and the sphere. You should already be familiar with its shape; if you have ever seen a Rubik's cube (the clue is in the name, right?), an ice cube (not the rapper), or a dice, you've seen a cube.

The takeaway from this section is that a cube is a 3D object; hence it has a volume . It's also highly regular, which means it's straightforward to find the volume of a cube.

Volume is a measure of the 3D space occupied by an object. But if you are not interested in abstract concepts and just want to know the volume of a cube, there is a simple answer to the question What is the volume of a cube?

volume = l³

where l is the length of the sides of the cube. This is just another way to say that you need to multiply the length of each side l by itself three times: l × l × l = l³ , or, in other words, elevating it to the third power (learn more about power in the exponent calculator )

The previous formula comes from the fact that the cube volume (in 3D) is analogous to the area of a square (in 2D). Like how you calculate the area of a square by multiplying the length of each side , you can multiply the three sides of a cube since they are all the same.

If all this sounds very easy to you, just know that there are other formulas for the volume of a cube in case you don't know the length of the sides . These are more complicated and will probably make you happier. If you are happy enough with the current difficulty level, let's move on.

Now that we have seen and understood the cube volume formula, we shall move on to explaining how to calculate the volume of a cube. We will first calculate the volume of a cube by hand, and later we will use the Omni-Calculator to find the volume of a cube without having to deal with the formula at all.

In true dad style , we will teach you how to do things the old-fashioned way before you move into the future. There is a good reason for this; it will help you better understand how you calculate the volume of a cube. Let's bring back the formula and use it in a simple example : volume = l³ . Assume we have a cube of side length l = 5 cm . The units don't really matter, but we'll keep them to help us keep track of the dimensions.

Take a piece of paper and proceed to attack the formula for the volume of a cube by multiplying first l × l = 5cm × 5cm = 25cm² . We have now calculated the area of the squares that make up each of the six sides of our cube. We are one dimension (i.e., one multiplication) away from finding the volume of a cube , so just pick up that pen again and let's do it!

volume = l³ = l² × l = 25cm² × 5cm = 125cm³ . And with that, we've got it - we have calculated the volume of a cube and escaped unharmed. Congratulations!

Now let us tell you a secret about a tool that lives to the left of this text and allows you to calculate the volume of a cube in one simple step. What do you say? Do you want to know more? Sure!

How to use Omni's volume of a cube calculator?

This is what you came here for. A calculator to solve all of your cube volume needs: Omni's volume of a cube calculator . Here at Omni, we have prepared a simple calculator that uses the formula for the volume of a cube to automatically compute the volume without any effort on your part.

All you need to do is to input the length of the side in the field named Side , and it will automatically compute the volume of a cube. It has never been so easy to answer the question: What is the volume of a cube? Alternatively, you can also calculate the length of a side of a cube if you already know its volume. Simply input the volume into the corresponding field and watch the magic happen (it's actually maths, but magic sounds cooler).

The calculator also does the reverse calculation in much the same way you would do it yourself. Take the formula for the volume of a cube and flip it around: volume = l³ => l = ³√volume , where ³√ is the cube root.

In case you haven't noticed, this Omni calculator has an "Advanced Mode." It extends the calculator's functionality, allowing you to calculate the volume of a cube from something other than the length of its sides. You can input the surface area, the face diagonal, or the cube diagonal.

The difference between the face diagonal and the cube diagonal might not be totally clear, so let's explain it a bit more. The cube diagonal is the three-dimensional distance between any two opposed corners. This is the largest distance between any two corners of a cube. When talking about the face diagonal, we are talking about the two-dimensional distance between the two furthermost corners of any of the squares that make up the six faces of a cube. All face diagonals are the same length.

As promised, we will now look at why the formula for the volume of a cube is so simple and why it is comprised of only two variables and two mathematical symbols. The main reason we could point to is the simplicity of the cube. The cube is highly regular and, most importantly, very easy to define. If you think about it, a sphere or a tetrahedron are even more regular than a cube, but it is much harder to compute their volume or area. This could be attributed partly to the fact that it is difficult to mathematically model the surface of a sphere when using the typical Cartesian coordinates.

The cube, however, follows precisely this pattern. The sides of a cube are always aligned with the unit vectors that generate the 3D Cartesian space. This makes the volume computation as easy as calculating the cross product of the three unitary Cartesian vectors (vector product), each multiplied by the length of the cube's sides .

You can see by checking the mathematical formulas that there seems to be a preference for squared shapes over rounded ones. If you don't believe it, take a look at the shapes of the rectangular prism calculator and the cylinder volume calculator and tell me which one you would prefer to compute by hand.

The preference for cubed shapes probably comes from their ease of construction and, more importantly, packing properties. As with squares and hexagons in 2D space, cubes can completely fill 3D spaces on their own when properly stacked. It might not sound all that special, but there are very few shapes with the ability to fill a space without leaving any gaps between them.

The advantage? Efficiency. If you make containers in the shape of cubes (rectangular prisms also work), you can be sure that you will be using all the space available, and you won't leave any dead space between them. It is this property alone that decides the shapes of containers, drawers, and wardrobes.

This is the reason ice cubes are cubes and not spheres, despite the latter being more energy-efficient. What we cannot really find a good answer for is why both rappers and physicists seem to love the name "ice-cube" but cannot agree on the spelling. Nature works in mysterious ways .

Cylinder volume

Meat footprint, square pyramid.

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• Calculating Volume and Area

How to Calculate the Volume of a Cube

Last Updated: March 16, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,712,462 times.

A cube is a three-dimensional shape that has equal width, height, and length measurements. A cube has six square faces, all of which have sides of equal length and all of which meet at right angles. [1] X Research source Finding the volume of a cube is a snap - generally, all that's needed is to multiply the cube's length × width × height . Since a cube's sides are all equal in length, another way of thinking of a cube's volume is s 3 , where s is the length of one of the cube's sides. See Step 1 below for a detailed breakdown of these processes.

Help Finding Volume of a Cube

Cubing One of the Cube's Sides

• To better understand the process of finding the volume of a cube, let's follow along with an example problem as we go through the steps in this section. Let's say the side of the cube is 2 inches (5.08 cm) long. We'll use this information to find the volume of the cube in the next step.

• This process is essentially the same as finding the area of the base and then multiplying it by the cube's height (or, in other words, length × width × height), since the area of the base is found by multiplying its length and its width. Since the length, width, and height of a cube are equal, we can shorten this process by simply cubing any of these measurements.
• Let's proceed with our example. Since the length of the side of our cube is 2 inches, we can find the volume by multiplying 2 x 2 x 2 (or 2 3 ) = 8 .

• In our example, since our original measurement was in inches, our final answer will be labelled with the units "cubic inches" (or in 3 ). So, our answer of 8 becomes 8 in 3 .
• If we had used a different initial unit of measurement, our final cubic units would differ. For instance, if our cube had sides with lengths of 2 meters , rather than 2 inches, we would label it with cubic meters (m 3 ).

Finding Volume from Surface Area

• The surface area of a cube is given via the formula 6 s 2 , where s is the length of one of the cube's sides. This formula is essentially the same as finding the 2-dimensional area of the cube's six faces and adding these values together. We'll use this formula to find the volume of the cube from its surface area. [7] X Research source
• As a running example, let's say that we have a cube whose surface we know to be 50 cm 2 , but we don't know its side lengths. In the next few steps, we'll use this information to find the cube's volume.

• In our example, dividing 50/6 = 8.33 cm 2 . Don't forget that two-dimensional answers have square units (cm 2 , in 2 , and so on).

• In our example, √8.33 is roughly 2.89 cm .

• In our example, 2.89 × 2.89 × 2.89 = 24.14 cm 3 . Don't forget to label your answer with cubic units.

Finding Volume from Diagonals

• For instance, let's say that one of a cube's faces has a diagonal that is 7 feet long. We would find the side length of the cube by dividing 7/√2 = 4.96 feet. Now that we know the side length, we can find the volume of the cube by multiplying 4.96 3 = 122.36 feet 3 .
• Note that, in general terms, d 2 = 2 s 2 where d is the length of the diagonal of one of the cube's faces and s is the length of one of the sides of the cube. This is because, according to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sums of the squares of the other two sides. Thus, because the diagonal of a cube's face and two of the sides on that face form a right triangle, d 2 = s 2 + s 2 = 2 s 2 .

• This is because of the Pythagorean Theorem. D , d , and s form a right triangle with D as the hypotenuse, so we can say that D 2 = d 2 + s 2 . Since we calculated above that d 2 = 2 s 2 , we can say that D 2 = 2 s 2 + s 2 = 3 s 2 .
• D 2 = 3 s 2 .
• 10 2 = 3 s 2 .
• 100 = 3 s 2
• 33.33 = s 2
• 5.77 m = s. From here, all we need to do to find the volume of the cube is to cube the side length.
• 5.77 3 = 192.45 m 3

Community Q&A

You Might Also Like

• ↑ https://www.mathopenref.com/cube.html
• ↑ https://www.gigacalculator.com/calculators/volume-calculator.php
• ↑ https://www.mathopenref.com/cubevolume.html
• ↑ https://www.cuemath.com/measurement/surface-area-of-cube/
• ↑ https://www.softschools.com/math/geometry/topics/surface_area_of_a_cube/
• ↑ https://www.omnicalculator.com/math/rectangular-prism#how-do-i-calculate-the-diagonal-of-a-rectangular-prism
• ↑ https://www.cuemath.com/measurement/volume-of-cube/

About This Article

To calculate the volume of a cube, find the length of one of the sides of the cube. When you have this measurement, multiply it by itself 2 times to get the volume, which is called “cubing” the number. For example, if your cube has a length of 2, you would multiply 2 × 2 × 2 to get a volume of 8. Be sure to include the units cubed with your answer. To learn more, like how to find the volume if you have the cube's surface area, keep reading! Did this summary help you? Yes No

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9.9: Solve Geometry Applications- Volume and Surface Area (Part 1)

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• Page ID 5010

Learning Objectives

• Find volume and surface area of rectangular solids
• Find volume and surface area of spheres
• Find volume and surface area of cylinders
• Find volume of cones

be prepared!

Before you get started, take this readiness quiz.

• Evaluate x 3 when x = 5. If you missed this problem, review Example 2.3.3 .
• Evaluate 2 x when x = 5. If you missed this problem, review Example 2.3.4 .
• Find the area of a circle with radius \(\dfrac{7}{2}\). If you missed this problem, review Example 5.6.12 .

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Problem Solving Strategy for Geometry Applications

• Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem and make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure \(\PageIndex{1}\)). The amount of paint needed to cover the outside of each box is the surface area , a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Figure \(\PageIndex{1}\) - This wooden crate is in the shape of a rectangular solid.

Each crate is in the shape of a rectangular solid . Its dimensions are the length, width, and height. The rectangular solid shown in Figure \(\PageIndex{2}\) has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Figure \(\PageIndex{2}\) - Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 4 by 2 by 3 rectangular solid has 24 cubic units.

Altogether there are 24 cubic units. Notice that 24 is the length × width × height.

The volume, V, of any rectangular solid is the product of the length, width, and height.

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B, is equal to length × width.

\[B = L \cdot W\]

We can substitute B for L • W in the volume formula to get another form of the volume formula.

\[\begin{split} V &= \textcolor{red}{L \cdot W} \cdot H \\ V &= \textcolor{red}{(L \cdot W)} \cdot H \\ V &= \textcolor{red}{B} h \end{split}\]

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4 × 2 × 3 rectangular solid we started with. See Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\)

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

\[\begin{split} A_{front} &= L \times W \qquad A_{side} = L \times W \qquad A_{top} = L \times W \\ A_{front} &= 4 \cdot 3 \qquad \quad \; A_{side} = 2 \cdot 3 \qquad \quad \; A_{top} = 4 \cdot 2 \\ A_{front} &= 12 \qquad \qquad A_{side} = 6 \qquad \qquad \; \; A_{top} = 8 \end{split}\]

Notice for each of the three faces you see, there is an identical opposite face that does not show.

\[\begin{split} S &= (front + back)+(left\; side + right\; side) + (top + bottom) \\ S &= (2 \cdot front) + (2 \cdot left\; side) + (2 \cdot top) \\ S &= 2 \cdot 12 + 2 \cdot 6 + 2 \cdot 8 \\ S &= 24 + 12 + 16 \\ S &= 52\; sq.\; units \end{split}\]

The surface area S of the rectangular solid shown in Figure \(\PageIndex{3}\) is 52 square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure \(\PageIndex{4}\)). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

\[S = 2LH + 2LW + 2WH\]

Figure \(\PageIndex{4}\) - For each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all.

Definition: Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length L, width W, and height H:

Example \(\PageIndex{1}\):

For a rectangular solid with length 14 cm, height 17 cm, and width 9 cm, find the (a) volume and (b) surface area.

Step 1 is the same for both (a) and (b), so we will show it just once.

Exercise \(\PageIndex{1}\):

Find the (a) volume and (b) surface area of rectangular solid with the: length 8 feet, width 9 feet, and height 11 feet.

Exercise \(\PageIndex{2}\):

Find the (a) volume and (b) surface area of rectangular solid with the: length 15 feet, width 12 feet, and height 8 feet.

1,440 cu. ft

Example \(\PageIndex{2}\):

A rectangular crate has a length of 30 inches, width of 25 inches, and height of 20 inches. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{3}\):

A rectangular box has length 9 feet, width 4 feet, and height 6 feet. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{4}\):

A rectangular suitcase has length 22 inches, width 14 inches, and height 9 inches. Find its (a) volume and (b) surface area.

2,772 cu. in

1,264 sq. in.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

\[\begin{split} V &= LWH \quad \; S = 2LH + 2LW + 2WH \\ V &= s \cdot s \cdot s \quad S = 2s \cdot s + 2s \cdot s + 2s \cdot s \\ V &= s^{3} \qquad \quad S = 2s^{2} + 2s^{2} + 2s^{2} \\ &\qquad \qquad \quad \; S = 6s^{2} \end{split}\]

So for a cube, the formulas for volume and surface area are V = s 3 and S = 6s 2 .

Definition: Volume and Surface Area of a Cube

For any cube with sides of length s,

Example \(\PageIndex{3}\):

A cube is 2.5 inches on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{5}\):

For a cube with side 4.5 meters, find the (a) volume and (b) surface area of the cube.

91.125 cu. m

121.5 sq. m

Exercise \(\PageIndex{6}\):

For a cube with side 7.3 yards, find the (a) volume and (b) surface area of the cube.

389.017 cu. yd.

319.74 sq. yd.

Example \(\PageIndex{4}\):

A notepad cube measures 2 inches on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{7}\):

A packing box is a cube measuring 4 feet on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{8}\):

A wall is made up of cube-shaped bricks. Each cube is 16 inches on each side. Find the (a) volume and (b) surface area of each cube.

4,096 cu. in.

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

• Math Article

Volume Of A Cube

Volume of a cube is the total cubic units occupied by it, in a three-dimensional space. A cube is a 3d-shape, that has six faces, twelve edges and eight vertices. Hence, the volume of a cube is the space enclosed by its six faces. Unlike, 2d shapes , it has additional dimensions apart from length and width, which is called height or thickness.  Therefore, the volume of cube is equal to product of its length, width and height. It is measured in cubic units. The more the value of its dimensions the more is the volume of cube.

In this article, we will discuss the definition and formula of volume of a cube along with examples.

What is the Volume of a Cube?

Definition: The volume of a cube defines the number of cubic units, occupied by the cube completely. The unit of volume of the cube is in cubic units such as centimeters 3 , meters 3 , inches 3 , feet 3 , etc.

A cube is a solid three-dimensional figure, which has 6 square faces or sides. The volume of a cube will be the total space occupied by it. Since all the faces of the cube are square in the shape, hence, the length of edges will also be equal. Therefore, the length, width and height of the cube are of equal length.

If length, width and height of cube is equal to ‘a’, then;

Volume of cube = a  × a × a

If we know the edge length i.e. “a”, then we can find the volume of the cube. Let us learn how to find the volume of any cubical structure.

Volume of a Cube Formula

We can easily find the volume of the cube (V), by knowing the length of its edges. Suppose, the length of the edges of the cube is ‘a’. Then the V will be the product of length, height, and width. So, the volume of the cube formula is:

Volume of a Cube = Length × Width × Height

Volume = a × a × a

Volume = a 3

Where ‘a’ is the length of the side of the cube or edges.

Derivation for Volume of a Cube

The volume of an object is defined as the amount of space a solid occupies. We know that a cube is a 3-dimensional object whose all the sides i.e. length, breadth, and height are equal. Now for a cube, the volume derivation will be as follows:

• Consider a square sheet of paper.
• Now, the area that the square sheet will take up will be its surface area i.e. its length multiplied by its breadth.
• As the square will have equal length and breadth, the surface area will be “a 2 “.
• Now, a cube is made by stacking multiple square sheets on top of each other so that the height becomes “a” units. This gives the height or thickness of the cube as “a”.
• Now, it can be concluded that the overall area covered by the cube will be the area of the base multiplied by the height.
• So,  Volume of Cube = a 2  × a = a 3

How to Find Volume of a Cube?

There are two methods by which we can find the volume of a cube.

• Using Edge-length
• Using Diagonal

Both the methods are formula-based and easy to understand.

Volume of Cube Given the Edge-Length

By the formula of volume of a cube, we know that;

Volume = (Edge of the cube) 3

Therefore, if we know the edge length of the cube, we can easily find its volume.

Example: If edge-length is 3 cm, the volume of the cube;

Volume = 3 3 = 3 x 3 x 3 = 27 cu.cm.

Volume of a Cube Given the Diagonal

The volume of a cube whose diagonal (d), is given is given by:

Hence, by measuring the length of the diagonal of cube, we can evaluate the volume of cube.

Example: If length of diagonal of cube is 3 inches, then the volume will be:

Volume = √3 × d 3 /9 = √3 × 3 3 /9 = 3√3 cu.in.

Check:  Diagonal of a Cube Formula

Video Lesson on Volume of Cube

For more information on volumes of cubes and cuboid, watch the below video:.

Surface Area of a Cube

The surface area of the cube is the total area occupied by its outer surface. We can find the surface area of a cube, which is equal to the number of square units that cover the surface of the cube, completely. The general formula of surface area for a cube of side ‘a’, is given by;

Surface Area of Cube = 6a 2

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Solved Examples on Volume of Cube

Question 1: Find the volume of the cube, having the sides of length 7 cm.

Given, the length of the sides of the cube is 7 cm.

We know, Volume of a cube = (length of sides of the cube) 3

Therefore, Volume, V = (7 cm) 3

V = 343 cm 3

Question 2: Find the length of the edges of the cube, if its volume is equal to 125 cm 3 .

Given, Volume of the cube = 125 cm 3 .

Let the length of the edges is ‘a’.

We know, by the formula,

The volume of a cube = (length of edges of the cube) 3

Substituting the value, we get,

Or a = 3 √125

Or a = 5 cm

Therefore, the length of the cube is 5 cm.

Question.3 : What is the volume of cube, whose length of diagonal is 27 cm.

Solution : By the formula of volume of cube given the diagonal, we have;

Volume = √3 × d 3 /9

d = 27 cm (given)

Volume of cube = √3 × 27 3 /9

= √3 × (27 × 27 × 27)/9

= √3 × (27 × 27 × 3)

= 3788 (Approx)

Practice Questions

• What is the volume of cube if length of diagonal is 15 cm?

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Frequently Asked Questions on Volume of a Cube

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SOLVING WORD PROBLEMS INVOLVING VOLUME OF CUBES

Formula : 

Volume of a cube = a 3 cubic units

Problem 1 :

Find the volume of a cube each of whose side is (i) 5 cm (ii) 3.5 m (iii) 21 cm

Volume of cube = a 3

(ii) 3.5 m :

= 42.875 cm 3

(iii) 21 cm :

= 9261 cm 3

Problem 2 :

A cubical milk tank can hold 125000 liters of milk. Find the length of its side in meters.

Volume of cubical milk tank = 125000 liters

1000 liters = 1 m 3 ,

Volume of tank = 125000/1000

a 3  = 125

Problem 3 :

A metallic cube with side 15 cm is melted and formed into a cuboid. If the length and height of the cuboid is 25 cm and 9 cm respectively then find the with of the cuboid.

volume of cuboid = volume of cube

l x w x h = a 3

25 x w x 9 = 15 3

225w = 3375

Divide each side by 225.

Problem 4 :

The sides of two cubes A and B are in the ratio 3 : 5. If the volume of cube A is 729 cm 3 , find the volume of cube B. 

From the ratio 3 : 5, the sides of cubes A and B are 

Volume of cube A = 243 cm 3

(3x) 3 = 729

27x 3  = 729

Divide each side by 27.

x 3  = 27

x 3  = 3 3

Side of cube A = 3(3) = 9 cm

Side of cube B = 5(3) = 15 cm

Volume of cube B : 

= 3375 cm 3

Problem 5 :

If the sides of two cubes are in the ratio 4 : 7, find the ratio of their volumes. 

From the ratio 4 : 7, the sides of two cubes are 

Ratio of their volumes :

= (4x) 3 : (7x) 3

= 64x 3  : 343x 3

Divide both the terms of the ratio by x 3 .

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How to Calculate the Volume of a Cube

Understanding the principles of geometry is a fundamental skill for high school students. Among the essential geometric shapes, the cube holds a special place. Learning how to calculate its volume not only helps you ace math exams but also equips you with practical problem-solving skills. In this blog post, we will explore the applications of cube calculations, provide a step-by-step guide to calculating the volume of a cube, highlight common mistakes to avoid, and even test your knowledge with an interactive quiz. Let’s dive in!

Practical Applications of Cube Calculations

Before we delve into the calculations, it’s crucial to understand the real-world significance of cube calculations. From architecture to manufacturing and even everyday life, cube calculations find practical applications in various fields. For instance:

• Architects use cube calculations to determine the volume of rooms and design layouts effectively.
• Engineers rely on cube measurements to calculate the capacity of containers or storage units.
• Builders use cube calculations to estimate the amount of material required for construction projects.

These are just a few examples that highlight the importance of mastering cube calculations in real-life scenarios.

Step-by-Step Guide to Calculating the Volume of a Cube

To calculate the volume of a cube, follow these simple steps:

• Definition of a Cube: Understand that a cube is a three-dimensional shape with six equal square faces.
• Identifying the Measurements Needed: Measure the length of any side of the cube. Make sure to use the same unit of measurement for all sides.
• Formula for Calculating Volume: The volume of a cube is calculated by multiplying the length of one side by itself twice. The formula is V = s^3, where V represents volume and s represents the length of one side.
• Step-by-Step Calculation Process: Apply the formula by substituting the length of one side into the equation. Raise the value to the power of 3 to find the volume. For example, if the length of one side is 5 units, the calculation would be V = 5^3 = 125 cubic units.

By following these steps, you can confidently calculate the volume of any cube you encounter.

Common Mistakes to Avoid

While cube calculations may seem straightforward, there are some common errors that students often make. Here are a few mistakes to be mindful of:

• Confusing the length of one side with the total length of all sides.
• Forgetting to raise the side length to the power of 3 when calculating the volume.
• Using different units of measurement for the length of each side.

Being aware of these common mistakes will help you avoid errors and ensure accurate calculations.

Quiz Questions and Answers

Here are three quiz questions to test your understanding of how to calculate the volume of a cube. Try to answer them before checking the solutions provided.

• Question: If each side of a cube is 4 units in length, what is the volume of the cube? Answer: The volume of the cube is 64 cubic units. (Using the formula V = s^3, where s = 4, V = 4^3 = 64)
• Question: A cube has a volume of 27 cubic units. What is the length of one side of the cube? Answer: The length of one side of the cube is 3 units. (Since V = s^3, and V = 27, then s = 3 because 3^3 = 27)
• Question: You are given 3 cubes with side lengths of 2 units, 3 units, and 4 units respectively. Which cube has the largest volume and what is it? Answer: The cube with the side length of 4 units has the largest volume, which is 64 cubic units. (Comparing V = 2^3 = 8, V = 3^3 = 27, and V = 4^3 = 64)

Mastering cube calculations is not just about solving math problems; it’s about developing critical thinking skills and practical problem-solving abilities. Understanding the volume of a cube opens doors to various fields where three-dimensional measurements are crucial. Remember, practice makes perfect, so keep honing your skills and seek out opportunities to apply your knowledge in real-world scenarios. Geometry tutors can help you master cube calculations and gain confidence in your mathematical abilities. Happy calculating!

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Volume of a Cube Calculator

Use this cube volume calculator to easily calculate the volume of a cube, given its side in any metric: mm, cm, meters, km, inches, feet, yards, miles...

    Calculation results

Related calculators.

• Volume of a cube formula
• How to calculate the volume of a cube?
• Example: find the volume of a cube

    Volume of a cube formula

The volume of a cube can be calculated if you know its side length. The formula is then volume cube = side 3 . This is simply the length of the side multiplied by itself two times. Illustration below:

Measuring the side of the cube is easy. The result from the calculation, using our volume of a cube calculator or otherwise, will always be in the length unit used, cubed. So, if you measured it in inches, the result will be in cubic inches. If the length was in feet, the result will be in cubic feet, and so on for cubic yards, cubic miles, cubic millimeters, cubic centimeters, cubic meters. If you need to find the volume in a specific unit, then a unit conversion would be needed which is most easily done to the length of the side, before replacing it in the formula.

    How to calculate the volume of a cube?

This is one of the simplest bodies one can calculate the volume of. You only need a single measurement, then you multiply it by itself and by itself again, equivalent to raising it to the power of 3. Finally, adjust the unit to its cubic equivalent. Simple as that.

In practical situations, you rarely know in advance that something is a cube, so you might still need to measure at least three sides before you can confirm this. If you have a plan or engineering schematic in which the measurements are all given, your task is significantly easier.

    Example: find the volume of a cube

The only variable one needs to know to compute the volume of any cube is the length of one of its sides. Since all sides are equal, it does not matter which side is given exactly. For example, if the length of a side is 5 inches, using the volume equation results in 5 3 = 5 x 5 x 5 = 125 in 3 (cubic inches).

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Volume of a Cube Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/volume-of-cube-calculator.php URL [Accessed Date: 11 Apr, 2024].

     Geometry calculators

WORD PROBLEMS ON SURFACE AREA AND VOLUME OF CUBE

In Geometry, a  Cube  is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges.

Volume of cube = a 3

Lateral surface area = 4a 2

Total surface area = 6a 2

To find side length of cube from the diagonal, we use the formula

Side length = a √3

Problem 1 :

Three cubes are joined end to end forming a cuboid. If side of a cube is 2 cm, find the dimensions of the cuboid thus obtained.

Given side of a cube = 2 cm

After joining 3 cubes,

length of cuboid = 2 × 3 = 6 cm, 

Height = 2 cm and  Breadth = 2 cm

Dimension of the cuboid is 6 cm x 2 cm x 2 cm.

Problem 2 :

Find the lateral surface area of a cube, if its diagonal is √6 cm.

Given, diagonal of the cube =  √6 cm

Diagonal of the cube = √3 a

Lateral surface area of cube = 4a²

= 4 × (√2)²

Problem 3 :

Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a single cube whose diagonal is 12√3 cm. find the edges of the three cubes.

Diagonal of the single cube = 12√3 cm

√3 a = 12√3

Volume of the single cube = sum of the volumes of the metallic cubes

a³ = (3x)³ + (4x)³ + (5x)³

(12)³ = 27x³ + 64x³ + 125x³

1738 = 216x³

x³ = 1728/216

Now, the edge of the first cube = 3(2) = 6 cm

Edge of the second cube = 4(2) = 8 cm

Edge of the third cube = 5(2) = 10 cm

Therefore, the edges of the three cubes are 6 cm, 8 cm, and 10 cm.

Problem 4 :

Volume of a cube is 5832 m³. Find the cost of painting its total surface area at the rate of $3.50 per m².

Volume of a cube is 5832 m³

a³ = 5832 m³

a =  5832

Total surface area = 6 × a²

= 6 × 324 = 1944 m²

Cost of painting at 3.50 per m² = 1944 × 3.50

Hence, the cost of painting is $6804

Problem 5 :

The cube has a surface area of 216 dm². Calculate:

a)     The area of one wall,

b)     Edge length,

c)     Cube volume.

The cube has a surface area of 216 dm²

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