Surface Area of Cubes and Cuboids

What you will learn

• Find the surface area of a cube and a cuboid.
• Use a net to see every face.
• Solve wrapping and painting problems.

Before you start: you should know how to find the area of a rectangle and what a net is. See Area and Perimeter and Nets, Faces, Edges and Vertices.

The idea

Surface area is the total area of all the faces of a solid. It tells you how much wrapping paper covers a gift or how much paint covers a box. It is measured in square units, such as $\text{cm}^2$ or $\text{m}^2$. Do not confuse it with volume, which fills the inside and uses cubic units.

Cube

A cube has $6$ identical square faces. If the side is $a$, one face has area $a^2$, so

$SA = 6a^2.$

Cuboid

A cuboid has $3$ pairs of matching rectangles: top and bottom ($l \times w$), front and back ($l \times h$), and left and right ($w \times h$). Add one of each pair and double:

$SA = 2(lw + lh + wh).$

Worked examples

Example 1: Cube

Find the surface area of a cube with sides of $7$ cm.

1. One face: $7 \times 7 = 49$.
2. Six faces: $6 \times 49 = 294$.

Answer: $294 \text{ cm}^2$

Example 2: Cuboid

A box is $20$ cm long, $15$ cm wide and $10$ cm high. Find its surface area.

1. Top and bottom: $20 \times 15 = 300$.
2. Front and back: $20 \times 10 = 200$.
3. Left and right: $15 \times 10 = 150$.
4. $2 \times (300 + 200 + 150) = 2 \times 650 = 1300$.

Answer: $1300 \text{ cm}^2$

Example 3: Painting a room

A room is $5$ m long, $4$ m wide and $2.5$ m high. Find the area of the four walls and ceiling to paint.

1. Ceiling: $5 \times 4 = 20$.
2. Two long walls: $2 \times (5 \times 2.5) = 25$.
3. Two short walls: $2 \times (4 \times 2.5) = 20$.
4. Total: $20 + 25 + 20 = 65$.

Answer: $65 \text{ m}^2$

Tip: list the six faces before adding. For an open box or a room, leave out the faces that are missing.

Why it matters

Surface area decides how much paint, wrapping, tile or cladding a job needs. It also explains why small animals lose heat faster than large ones and why ice cubes melt faster when crushed: more surface for the same volume.