Finding an Edge of a Cube or Cuboid

Learn how to find unknown dimensions when given the volume

1. Volume Formula Basics

What is Volume?

Volume is the amount of space a solid takes up. We measure volume in cubic units like cm³ or m³.

Think of it as counting how many unit cubes can fit inside the solid!

Visualising Volume with Unit Cubes

A cuboid with length 4 cm, breadth 3 cm and height 2 cm is made of 4 × 3 × 2 = 24 unit cubes, each with edges of 1 cm.

Volume = 4 × 3 × 2 = 24 cm³

Key Formulas

Volume of Cube

Volume = Edge × Edge × Edge

Volume = Edge³

Volume of Cuboid

Volume = Length × Breadth × Height

Example 1: Find the Volume

Find the volume of a cuboid with length 8 m, breadth 4 m, and height 3 m.

Cuboid: 8 m × 4 m × 3 m

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Solution:

Volume of cuboid = Length × Breadth × Height

= 8 × 4 × 3

= 96 m³

The volume of the cuboid is 96 m³.

2. Finding an Unknown Dimension

Key Insight

Since Volume = Length × Breadth × Height, we can rearrange to find any unknown dimension:

Height = Volume ÷ (Length × Breadth)

Breadth = Volume ÷ (Length × Height)

Length = Volume ÷ (Breadth × Height)

Example 2: Finding the Height

A cuboid has a volume of 60 cm³. Its length is 5 cm and its breadth is 4 cm. Find its height.

Volume = 60 cm³. Find the height.

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Solution:

Volume of cuboid = 60 cm³

Height of cuboid = Volume ÷ (Length × Breadth)

$= \frac{60}{5 \times 4}$

$= \frac{60}{20}$

= 3 cm

The height of the cuboid is 3 cm.

Example 3: Finding the Length

Lucas has a box with a volume of 480 cm³. The height of the box is 6 cm and its breadth is 8 cm. What is the length of the box?

Volume = 480 cm³. Find the length. (Not drawn to scale.)

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Solution:

Volume of box = 480 cm³

Length of box = Volume ÷ (Breadth × Height)

$= \frac{480}{8 \times 6}$

$= \frac{480}{48}$

= 10 cm

The length of the box is 10 cm.

3. Finding the Edge of a Cube

Cube Root

For a cube, all edges are equal. If the volume is V, then:

Volume = Edge × Edge × Edge = Edge³

To find the edge, we need the cube root:

Edge of cube = $\sqrt[3]{\text{Volume}}$

The cube root symbol $\sqrt[3]{\phantom{x}}$ means "which number, written three times and multiplied together (as in 2 × 2 × 2), gives this value?"

Common Perfect Cubes

Edge	Volume (Edge³)	Cube Root
1	1	$\sqrt[3]{1} = 1$
2	8	$\sqrt[3]{8} = 2$
3	27	$\sqrt[3]{27} = 3$
4	64	$\sqrt[3]{64} = 4$
5	125	$\sqrt[3]{125} = 5$
6	216	$\sqrt[3]{216} = 6$
7	343	$\sqrt[3]{343} = 7$
8	512	$\sqrt[3]{512} = 8$
9	729	$\sqrt[3]{729} = 9$
10	1000	$\sqrt[3]{1000} = 10$

Example 4: Finding the Edge of a Cube

A cubical box has a volume of 729 cm³. What is the length of one edge of the box?

A cube with volume 729 cm³

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Solution:

Volume of cubical box = 729 cm³

For a cube: Volume = Edge × Edge × Edge

729 = 9 × 9 × 9

Length of one edge = $\sqrt[3]{729}$ = 9 cm

The length of one edge of the cubical box is 9 cm.

4. Using Face Area to Find Dimensions

Key Relationships

When you know the area of a face:

For Cuboid (when you also know the volume):

Volume = Base Area × Height

So: Height = Volume ÷ Base Area

For Cube:

Face Area = Edge × Edge = Edge²

So: Edge = $\sqrt{\text{Face Area}}$ (square root)

Example 5: Using Base Area

The area of the base of a cuboid is 70 cm² and its volume is 1050 cm³. Find the height of the cuboid.

Base Area = 70 cm², Volume = 1050 cm³. (Not drawn to scale.)

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Solution:

Volume = 1050 cm³

Since Volume = Base Area × Height:

Height of cuboid = Volume ÷ Base Area

$= \frac{1050}{70}$

= 15 cm

The height of the cuboid is 15 cm.

Example 6: Finding Edge from Face Area

The area of the shaded face of a cube is 36 cm². Find the length of one edge of the cube.

Face Area = 36 cm²

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Solution:

Area of square face = 36 cm²

Area of square face = Edge × Edge = Edge²

36 = 6 × 6

Length of one edge = $\sqrt{36}$ = 6 cm

The length of one edge of the cube is 6 cm.

Key Takeaways

1.Volume of cube = Edge × Edge × Edge = Edge³
2.Volume of cuboid = Length × Breadth × Height
3.To find unknown dimension: Divide volume by the product of the other two dimensions
4.Edge of cube = Cube root of volume ( $\sqrt[3]{V}$ )
5.Height from base area = Volume ÷ Base Area
6.Edge from face area (cube) = Square root of face area ( $\sqrt{A}$ )