5.4 Surface Area and Volume

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What Are We Measuring?

Before diving into each shape, it helps to understand two big ideas:

• Volume — how much space a 3D shape takes up. Think of it as how much water you could fill it with. Volume is measured in cubic units: cm³, m³, mm³.
• Surface area — the total area of all the outer faces (sides) of a 3D shape. Think of it as how much wrapping paper you would need to cover the outside completely. Surface area is measured in square units: cm², m², mm².

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Volume of a Cuboid

Where:

• $l$ = length
• $w$ = width
• $h$ = height

For a cube with side length $L$: $V = L^3$

Surface Area of a Cuboid

A cuboid has three pairs of identical rectangular faces:

• Top and bottom: each has area $l \times w$ → total $2(l \times w)$
• Front and back: each has area $l \times h$ → total $2(l \times h)$
• Left and right sides: each has area $w \times h$ → total $2(w \times h)$

$$\text{TSA} = 2(lw) + 2(lh) + 2(wh)$$

For a cube with side $L$: each face is a square with area $L^2$, and there are 6 faces:

$$\text{TSA} = 6L^2$$

Worked Example — Cuboid

A cuboid has length 6 cm, width 4 cm, and height 3 cm. Find its volume and total surface area.

Volume:

$$V = 6 \times 4 \times 3 = 72 \text{ cm}^3$$

Surface area:

$$\text{TSA} = 2(6 \times 4) + 2(6 \times 3) + 2(4 \times 3)$$

$= 2(24) + 2(18) + 2(12)$

$$= 48 + 36 + 24 = 108 \text{ cm}^2$$