Volume of Cubes

Learn to find the volume of a cube with V = s³, work with whole and decimal edges, find the edge from the volume and link cm³ to liters.

Elementary20 min18 problems
Practice Problems

Definition

A cube is a 3D shape with 6 identical square faces. All 12 of its edges have the same length, which we call ss.
The volume of a cube is the amount of space inside it. You already know that a rectangular prism has V=l×w×hV = l \times w \times h. A cube is the special case where length, width and height are all the same number ss:
V=s×s×s=s3V = s \times s \times s = s^3
We read s3s^3 as "ss cubed". Volume is always written in cubic units: if the edge is in cm, the volume is in cm³; if the edge is in m, the volume is in m³.
The numbers you get by cubing whole numbers are called perfect cubes (or cube numbers):
1, 8, 27, 64, 125, 216, 343, 512, 729, 10001,\ 8,\ 27,\ 64,\ 125,\ 216,\ 343,\ 512,\ 729,\ 1000
Going backwards, from the volume to the edge, is called taking the cube root: s=V3s = \sqrt[3]{V}. For example, 643=4\sqrt[3]{64} = 4 because 4×4×4=644 \times 4 \times 4 = 64.

Try it now

Which formula gives the volume of a cube with edge ss?

Worked Examples

A wooden block is a cube with an edge of 55 cm. What is its volume?
1

Write the cube formula

V=s3=s×s×sV = s^3 = s \times s \times s → Formula ready
2

Substitute the edge

V=5×5×5V = 5 \times 5 \times 5 → Value plugged in
3

Multiply the first two edges

5×5=255 \times 5 = 25 → Area of one face: 2525 cm²
4

Multiply by the third edge

25×5=12525 \times 5 = 125 → V=125V = 125 cm³

Common Mistakes

Multiplying the edge by 3: for s=5s = 5, writing V=5×3=15V = 5 \times 3 = 15

Why it's wrong: s3s^3 means ss multiplied by itself 3 times, not ss times 3.

Correct: 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125 cm³.

Stopping after two edges: 5×5=255 \times 5 = 25

Why it's wrong: s×ss \times s is the area of ONE face (square units). Volume needs all three dimensions.

Correct: Multiply by the edge one more time: 25×5=12525 \times 5 = 125 cm³, and write cubic units.

Thinking that doubling the edge doubles the volume

Why it's wrong: The edge appears three times in s×s×ss \times s \times s, so each of the three factors doubles.

Correct: Doubling the edge multiplies the volume by 23=82^3 = 8. Tripling it multiplies the volume by 33=273^3 = 27.

Misplacing the decimal point: 0.23=0.80.2^3 = 0.8 or 0.60.6

Why it's wrong: Each multiplication by a decimal adds decimal places: one decimal place three times gives three decimal places.

Correct: 0.2×0.2=0.040.2 \times 0.2 = 0.04, then 0.04×0.2=0.0080.04 \times 0.2 = 0.008. A small edge gives an even smaller volume.

Interactive Visual

3D Shape Viewer

Faces

6

Edges

12

Vertices

8

Face shape

Square

Faces at each vertex

3

Euler's formula: V − E + F = 8 − 12 + 6 = 2

Volume

V = s³

V = 5³ = 125

125 cm³

Surface Area

SA = 6s²

SA = 6 · 5² = 150

150 cm²

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which formula gives the volume of a cube with edge ss?

Why It Matters

Cubes are everywhere, and their volume is the easiest one to calculate:
  • Kitchen: an ice cube with a 3 cm edge holds 2727 cm³ of water
  • Liters: a cube with a 10 cm edge holds exactly 10001000 cm³, which is 1 liter. This is how the liter is defined!
  • Packing: shipping boxes, storage cubes and dice are all cubes
  • Building: concrete blocks, cubic tanks and cube-shaped rooms
  • Scaling: when a cube gets 2 times bigger on every edge, it holds 8 times as much. Knowing this helps you guess sizes and avoid costly mistakes.
The cube formula is also the first step toward cylinders, cones, spheres and pyramids, which you will meet in the next lessons.

Real World Applications

Ice Cube Trays

Ice cube trays make small cubes of ice. Knowing the volume of one cube tells you how much water a whole tray needs.

Example:

An ice cube with a 22 cm edge holds 23=82^3 = 8 cm³ of water.

1Try It Yourself

A tray makes 1212 ice cubes. Each cube has an edge of 33 cm.

How many cm³ of water do you need to fill the whole tray?

Step 1: Write the mathematical expression

Find the volume of one cube, then multiply by 12:

Storage Cubes

Fabric storage cubes for shelves come in cube shapes. Their volume in liters tells you how much they can hold.

Example:

A storage cube with a 1010 cm edge holds 10001000 cm³, which is exactly 11 liter.

2Try It Yourself

A storage cube has an edge of 4040 cm.

How many liters does it hold? (11 L =1000= 1000 cm³)

Step 1: Write the mathematical expression

Cube the edge, then divide by 1000:

Toy Building Blocks

Toy companies pack small cube blocks into larger cube boxes. The box must hold a whole number of blocks along each edge.

Example:

A box with a 66 cm edge holds 3×3×3=273 \times 3 \times 3 = 27 blocks with a 22 cm edge.

3Try It Yourself

A cube-shaped box has an inside edge of 1010 cm. It is filled with toy cubes that have a 22 cm edge.

How many toy cubes fit in the box?

Step 1: Write the mathematical expression

Count the blocks along one edge, then cube that number:

Key Takeaways

  • 1A cube has 6 equal square faces and 12 equal edges of length ss
  • 2Volume of a cube: V=s×s×s=s3V = s \times s \times s = s^3, the special case of l×w×hl \times w \times h
  • 3Always write cubic units: cm³, m³, in³
  • 4Perfect cubes to know: 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000
  • 5To find the edge from the volume, take the cube root: s=V3s = \sqrt[3]{V}
  • 6Doubling the edge multiplies the volume by 88; a cube with a 1010 cm edge holds 11 liter

Frequently Asked Questions

Because s×s×ss \times s \times s is exactly the volume of a cube with edge ss. The name of the power comes from the shape!
Because s×s×ss \times s \times s is exactly the volume of a cube with edge ss. The name of the power comes from the shape!
Yes. A cube is a rectangular prism where length, width and height are equal. That is why V=l×w×hV = l \times w \times h becomes V=s×s×sV = s \times s \times s.
Find the number that multiplied by itself 3 times gives the volume. That is the cube root: 1253=5\sqrt[3]{125} = 5 because 53=1255^3 = 125. Learning the perfect cubes up to 10001000 makes this fast.
The formula works the same way: (12)3=12×12×12=18\left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}. A cube with a half-unit edge has one eighth of a cubic unit of volume.

Glossary

Cube
A 3D shape with 6 identical square faces, 12 equal edges and 8 vertices
Edge
A line segment where two faces meet; in a cube all edges have the same length ss
Cubic unit
The volume of a cube with a 1-unit edge, such as 1 cm³ or 1 m³
Perfect cube
A number that is a whole number cubed, such as 27=3327 = 3^3 or 64=4364 = 4^3
Cube root
The number that, cubed, gives a value: 2163=6\sqrt[3]{216} = 6 because 63=2166^3 = 216

Formula Card

Volume of a cube

V=s3=s×s×sV = s^3 = s \times s \times s

Multiply the edge by itself three times

Edge from volume

s=V3s = \sqrt[3]{V}

Find the number that, cubed, gives the volume

Scaling the edge

(k⋅s)3=k3⋅s3(k \cdot s)^3 = k^3 \cdot s^3

Edge times 2 gives volume times 8; edge times 3 gives volume times 27

The liter cube

103=1000 cm3=1 L10^3 = 1000 \text{ cm}^3 = 1 \text{ L}

A cube with a 10 cm edge holds exactly 1 liter

Today's challengeOne problem, three tries

More in This Topic

Introduction to Volume

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Metric Volume Units (mL, L)

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Volume of Rectangular Prisms

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Imperial Volume Units (cups, pints, gallons)

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