Teacher Guide: 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.

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10 questions on Volume & Capacity. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Calculate the volume of a cube with V = s³ for whole-number and decimal edges
Explain why the cube formula is the special case of V = l × w × h
Find the edge of a cube from its volume for perfect cubes up to 1000
Describe how the volume changes when the edge is doubled or tripled
Convert the volume of a cube from cm³ to liters

Prerequisites

• Volume of rectangular prisms (V = l × w × h)
• Introduction to volume and cubic units
• Metric volume units (1 L = 1000 mL)
• Multiplying whole numbers and decimals
• Exponents (what 4³ means)

Discussion Starters

1. Why do you think we say "cubed" for the third power?
2. If you build a cube with 3 small cubes along each edge, how many small cubes do you need? What about 4?
3. A friend says doubling the edge of a box doubles what it can hold. How would you show them they are wrong?
4. Why is a cube with a 10 cm edge a good way to picture 1 liter?
5. Can a cube have a volume of 50 cm³? What would its edge be close to?

Common Misconceptions

s³ means 3 × s

Remediation: Build a 2 × 2 × 2 cube from unit cubes: it uses 8 cubes, not 6. Write 2³ = 2 × 2 × 2 next to the model.

Doubling the edge doubles the volume

Remediation: Show that a cube with a 2-unit edge takes exactly 8 unit cubes. Ask students to predict, then build, the cube with a 4-unit edge (64 cubes vs 8).

Cubing a decimal makes it bigger

Remediation: Compare 0.5³ = 0.125 with 0.5 on a number line. Multiplying by a number less than 1 makes the result smaller.

Differentiation Ideas

For Struggling Students:

• Use unit cubes to build and count small cubes before using the formula
• Keep a reference card with the perfect cubes 1 to 1000
• Start with edges from 1 to 5 before moving to larger numbers

For On-Level Students:

• Calculate volumes with decimal edges such as 1.5 and 2.5
• Find the edge from a perfect cube volume
• Convert cube volumes from cm³ to liters

For Advanced Students:

• Estimate the edge of a cube with a volume that is not a perfect cube, like 100 cm³
• Investigate how surface area and volume change differently when the edge doubles
• Use the calculator to check fractional edges such as (1/2)³ and (3/2)³

Standards Alignment

5.MD.C.5 (CCSS.MATH.CONTENT.5.MD.C.5)
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume
6.G.A.2 (CCSS.MATH.CONTENT.6.G.A.2)
Find the volume of a right rectangular prism with fractional edge lengths
8.EE.A.2 (CCSS.MATH.CONTENT.8.EE.A.2)
Evaluate square roots of small perfect squares and cube roots of small perfect cubes

Lesson Resources

visual3D Cube Viewer
Change the edge of a cube and watch the volume formula update
activityBuild the Perfect Cubes
Students build cubes with 1, 2, 3 and 4 unit cubes per edge and record 1, 8, 27, 64
activityThe Liter Cube
Fill a 10 cm cube container with water and pour it into a 1 liter jug
worksheetCube Volume Practice
Whole and decimal edges, edges from volumes and scaling problems

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A cube is a 3D shape with 6 identical square faces. All 12 of its edges have the same length, which we call

s

The volume of a cube is the amount of space inside it. You already know that a rectangular prism has

V = l \times w \times h

. A cube is the special case where length, width and height are all the same number

s

V = s \times s \times s = s^3

We read

s^3

as "

s

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,\ 1000

Going backwards, from the volume to the edge, is called taking the cube root:

s = \sqrt[3]{V}

. For example,

\sqrt[3]{64} = 4

because

4 \times 4 \times 4 = 64

Worked Examples

A wooden block is a cube with an edge of $5$ cm. What is its volume?

Write the cube formula

$V = s^3 = s \times s \times s$ → Formula ready

Substitute the edge

$V = 5 \times 5 \times 5$ → Value plugged in

Multiply the first two edges

$5 \times 5 = 25$ → Area of one face: $25$ cm²

Multiply by the third edge

$25 \times 5 = 125$ → $V = 125$ cm³

Answer: The volume of the block is $125$ cm³.

Common Mistakes

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

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

Correct: $5^3 = 5 \times 5 \times 5 = 125$ cm³.

Stopping after two edges: $5 \times 5 = 25$

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

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

Thinking that doubling the edge doubles the volume

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

Correct: Doubling the edge multiplies the volume by $2^3 = 8$ . Tripling it multiplies the volume by $3^3 = 27$ .

Misplacing the decimal point: $0.2^3 = 0.8$ or $0.6$

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

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

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 $27$ cm³ of water
Liters: a cube with a 10 cm edge holds exactly $1000$ 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 $2$ cm edge holds $2^3 = 8$ cm³ of water.

1Try It Yourself

A tray makes $12$ ice cubes. Each cube has an edge of $3$ 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 $10$ cm edge holds $1000$ cm³, which is exactly $1$ liter.

2Try It Yourself

A storage cube has an edge of $40$ cm.

How many liters does it hold? ( $1$ L $= 1000$ 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 $6$ cm edge holds $3 \times 3 \times 3 = 27$ blocks with a $2$ cm edge.

3Try It Yourself

A cube-shaped box has an inside edge of $10$ cm. It is filled with toy cubes that have a $2$ 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 $s$
2Volume of a cube: $V = s \times s \times s = s^3$ , the special case of $l \times w \times h$
3Always write cubic units: cm³, m³, in³
4Perfect cubes to know: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$
5To find the edge from the volume, take the cube root: $s = \sqrt[3]{V}$
6Doubling the edge multiplies the volume by $8$ ; a cube with a $10$ cm edge holds $1$ liter

Frequently Asked Questions

Why is $s^3$ called "s cubed"?

Because

s \times s \times s

is exactly the volume of a cube with edge

s

. The name of the power comes from the shape!

Is a cube a rectangular prism?

Yes. A cube is a rectangular prism where length, width and height are equal. That is why

V = l \times w \times h

becomes

V = s \times s \times s

How do I find the edge if I only know the volume?

Find the number that multiplied by itself 3 times gives the volume. That is the cube root:

\sqrt[3]{125} = 5

because

5^3 = 125

. Learning the perfect cubes up to

1000

makes this fast.

What if the edge is a fraction?

The formula works the same way:

\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 $s$
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 $27 = 3^3$ or $64 = 4^3$
Cube root: The number that, cubed, gives a value: $\sqrt[3]{216} = 6$ because $6^3 = 216$

Formula Card

Volume of a cube

V = s^3 = s \times s \times s

Multiply the edge by itself three times

Edge from volume

s = \sqrt[3]{V}

Find the number that, cubed, gives the volume

Scaling the edge

(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

10^3 = 1000 \text{ cm}^3 = 1 \text{ L}

A cube with a 10 cm edge holds exactly 1 liter

Teacher Guide: Volume of Cubes

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Ice Cube Trays

Storage Cubes

Toy Building Blocks

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Volume of a cube

Edge from volume

Scaling the edge

The liter cube

More in This Topic

Introduction to Volume

Metric Volume Units (mL, L)

Volume of Rectangular Prisms

Imperial Volume Units (cups, pints, gallons)