Volume and Surface Area of a Cuboid

Formulas for the volume and surface area of a cuboid (rectangular box) with worked examples.

The Cuboid

A cuboid (rectangular box) has six rectangular faces. Its cube-shaped cousin is a special case where $l = w = h$ .

V = l \cdot w \cdot h

Length × width × height.

SA = 2(lw + lh + wh)

Two of each face: top/bottom, front/back, left/right.

Cuboid: l = 5 cm, w = 3 cm, h = 4 cm

Volume: $V = 5 \times 3 \times 4 = 60 \text{ cm}^3$

Surface area: $SA = 2(5\times3 + 5\times4 + 3\times4) = 2(15 + 20 + 12) = 2 \times 47 = 94 \text{ cm}^2$

Real world: swimming pool volume

A rectangular swimming pool is 10 m long, 5 m wide, and 1.8 m deep. The volume of water needed to fill it:

$V = 10 \times 5 \times 1.8 = 90 \text{ m}^3$

Since 1 m³ = 1,000 litres, the pool holds 90,000 litres — useful for calculating water bills and chemical dosing.

Real world: how many removal boxes to order

A removal box measures 60 cm × 40 cm × 30 cm. The household has a room of roughly 18 m³ of contents.

Each box holds: $V = 0.6 \times 0.4 \times 0.3 = 0.072 \text{ m}^3$

Boxes needed: $18 \div 0.072 = 250$ boxes (in practice fewer, since not everything is box-shaped).

Real world: brickwork estimation

A standard house brick measures 21.5 cm × 10.25 cm × 6.5 cm. Volume of one brick:

$V = 21.5 \times 10.25 \times 6.5 \approx 1{,}432 \text{ cm}^3$

A builder calculating how many bricks to order for a wall uses this volume to also estimate the total weight (density of brick ≈ 1.9 g/cm³).

Tip

Tip: To convert volumes — 1 m³ = 1,000,000 cm³ = 1,000 litres. Always convert all lengths to the same unit before calculating.

A box is 8 cm long, 6 cm wide, and 2 cm high. Find its volume and surface area.

Show answer

V = 8 × 6 × 2 = 96 cm³. SA = 2(48 + 16 + 12) = 2 × 76 = 152 cm².