Finding the Area of One Face

Learn to find face areas using volume and other dimensions

1. Finding Base Area from Volume

Key Relationship

We know that: Volume = Length × Breadth × Height

Since Length × Breadth = Base Area, we can write:

Volume = Base Area × Height

Base Area = Volume ÷ Height

Understanding the Relationship

Think of stacking layers. Each layer has an area equal to the base. The volume is the base area times the number of layers (height).

Base area = 5 × 4 = 20 cm². With 3 layers, Volume = 20 × 3 = 60 cm³

Example 1: Finding Base Area

A rectangular container has a volume of 960 m³ and a height of 8 m. Find the base area of the container.

Volume = 960 m³, Height = 8 m (not drawn to scale)

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Solution:

Volume = Base Area × Height

Base Area = Volume ÷ Height

= 960 ÷ 8

= 120 m²

The base area of the container is 120 m².

Practice: Try This!

A cuboid has a volume of 750 cm³ and a height of 5 cm. Find its base area.

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Base Area = Volume ÷ Height = 750 ÷ 5 = 150 cm²

2. Finding Face Area of a Cuboid

Different Faces of a Cuboid

A cuboid has 6 faces. We can find the area of any face by dividing the volume by the length of the edge at right angles to that face:

Top/Bottom Face

Area = Length × Breadth

= Volume ÷ Height

Front/Back Face

Area = Length × Height

= Volume ÷ Breadth

Left/Right Face

Area = Breadth × Height

= Volume ÷ Length

Example 2: Finding a Shaded Face Area

The volume of a cuboid is 3840 cm³. The breadth of the cuboid is 10 cm. What is the area of the shaded face?

Volume = 3840 cm³, Breadth = 10 cm. Find the shaded face area. (Not drawn to scale.)

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Solution:

Volume = Length × Breadth × Height = 3840 cm³

We can write: Volume = Breadth × Area of shaded face

Area of shaded face = Volume ÷ Breadth

= 3840 ÷ 10

= 384 cm²

The area of the shaded face is 384 cm².

Example 3: Finding a Length from a Face Area

The volume of a cuboid is 1190 cm³. The area of its shaded face is 85 cm². Find the length of the cuboid.

Volume = 1190 cm³, Shaded face area = 85 cm²

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Solution:

Volume = Length × Area of shaded face

Length of cuboid = Volume ÷ Area of shaded face

$= \frac{1190}{85}$

= 14 cm

The length of the cuboid is 14 cm.

3. Finding Face Area of a Cube

Two-Step Process

To find the face area of a cube when given its volume:

Step 1: Find the edge length using cube root

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

Step 2: Calculate the face area

Face Area = Edge × Edge = Edge²

Example 4: Face Area of a Cube

The volume of a cube is 5832 cm³. Find the area of a square face of the cube.

A cube with volume 5832 cm³. Find the area of the shaded face.

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Solution:

Volume = 5832 cm³

Step 1: Find the edge length

Edge = $\sqrt[3]{5832}$

Since 18 × 18 × 18 = 5832

Edge = 18 cm

Step 2: Find the face area

Area of a square face = Edge × Edge

= 18 × 18

= 324 cm²

The area of a square face of the cube is 324 cm².

Larger Perfect Cubes (Useful Reference)

Edge	Volume (Edge³)	Face Area (Edge²)
11	1331	121
12	1728	144
13	2197	169
14	2744	196
15	3375	225
18	5832	324
20	8000	400

Key Takeaways

1.Base Area = Volume ÷ Height
2.Face Area of Cuboid = Volume ÷ (length of the edge at right angles to that face)
3.For Cube Face Area: First find edge ( $\sqrt[3]{V}$ ), then square it (Edge²)
4.Remember: Volume has cubic units (cm³), Area has square units (cm²)
5.Use the calculator to find the cube root of a large number