Surface Area and Volume of a Cube

The Cube

A cube is a three-dimensional shape with six equal square faces.

Volume of a Cube

V = a^3

Volume is the edge length cubed.

Surface Area of a Cube

S = 6a^2

A cube has 6 faces, each with area a².

Rectangular Box (Cuboid)

For a box with dimensions $l \times w \times h$ :

Volume of a Cuboid

V = l \cdot w \cdot h

Surface Area of a Cuboid

S = 2(lw + lh + wh)

Cube with edge 4 cm

$V = 4^3 = 64 \text{ cm}^3$

$S = 6 \times 4^2 = 6 \times 16 = 96 \text{ cm}^2$

Real-World Applications

Real world: estimating cardboard for a cereal box

A cereal box measures 30 cm × 8 cm × 20 cm. The surface area tells us how much cardboard is needed:

$SA = 2(30 \times 8 + 30 \times 20 + 8 \times 20) = 2(240 + 600 + 160) = 2 \times 1{,}000 = 2{,}000 \text{ cm}^2$

The volume tells how much cereal fits inside: $V = 30 \times 8 \times 20 = 4{,}800 \text{ cm}^3$ .

Real world: dice manufacturing

A gaming die has edge length 1.6 cm. The surface area determines the amount of paint (one colour per face):

$S = 6 \times 1.6^2 = 6 \times 2.56 = 15.36 \text{ cm}^2$

A factory producing 100,000 dice needs enough paint to cover $1{,}536{,}000 \text{ cm}^2$ — about 154 m².

Real world: wrapping a gift

A square gift box has edge 25 cm. The surface area gives the minimum wrapping paper needed:

$S = 6 \times 25^2 = 6 \times 625 = 3{,}750 \text{ cm}^2 = 0.375 \text{ m}^2$

In practice you add 10–15% extra for overlaps and folds.

Tip

Surface area = total outside area (painting, wrapping, labels). Volume = space inside (filling, storing). Always check which one the question is asking for.

Practice Problems

Find the volume and surface area of a cube with edge 3 m.

Show answer

V = 27 m³, S = 6 × 9 = 54 m²

A box is 5 cm × 3 cm × 2 cm. Find its volume and surface area.

Show answer

V = 30 cm³, S = 2(15 + 10 + 6) = 62 cm²