Surface Area and Volume

Two questions about every solid

Whenever you meet a 3D (solid) shape there are two natural questions to ask:

1. How much paper would I need to cover it completely? That is the surface area.
2. How much could I fit inside it? That is the volume.

These ideas are everywhere: the cardboard in a box, the paint on a wall, the water in a tank, the air in a room. This lesson shows you how to work out both for the most common solids. If you are still building confidence with flat shapes, review area and perimeter first, because surface area is really just lots of areas added together.

Surface area: covering the outside

The surface area of a solid is the total area of all its faces (its flat outside surfaces). Because it is built from areas, it is always measured in square units — cm², m² and so on.

The reliable method is:

Find the area of every face, then add them all up.

Worked example 1 — surface area of a cuboid

A cuboid (a box shape) measures 6 cm long, 4 cm wide, 3 cm high. Find its surface area.

A cuboid has 6 faces that come in 3 matching pairs:

Face	Dimensions	Area	Pairs
Front & back	6 × 3	18 cm²	× 2 = 36
Top & bottom	6 × 4	24 cm²	× 2 = 48
Left & right	4 × 3	12 cm²	× 2 = 24

Add the three pairs: 36 + 48 + 24 = 108 cm².

There is a handy formula for a cuboid: SA = 2(lw + lh + wh). Check: 2(24 + 18 + 12) = 2 × 54 = 108 cm². The formula and the face-by-face method always agree, because the formula simply does the same additions for you.

Worked example 2 — surface area of a cube

A cube has 6 identical square faces. For a cube with edge length 5 cm:

Step 1 — Area of one face: 5 × 5 = 25 cm² Step 2 — Multiply by 6 faces: 6 × 25 = 150 cm²

So SA of a cube = 6 × (edge)².

Volume: filling the inside

The volume of a solid is the amount of space inside it. Picture filling the shape with tiny unit cubes, each 1 cm on every side. The number of cubes that fit is the volume in cubic centimetres (cm³). Because we multiply three lengths, the units are cubic.

Worked example 3 — volume of a cuboid

For the same box (6 cm × 4 cm × 3 cm):

Volume of a cuboid = length × width × height

Step 1: 6 × 4 = 24 (this is the number of cubes in one layer along the base) Step 2: 24 × 3 = 72 (we stack 3 such layers)

The volume is 72 cm³. Thinking of it as layers explains why the formula works: you find how many cubes cover the base, then multiply by how many layers high the shape is.

Worked example 4 — volume of a cube

For a cube with edge 5 cm: Volume = 5 × 5 × 5 = 125 cm³, so volume of a cube = (edge)³.

The big idea: every prism follows one rule

A prism is a solid with the same cross-section all the way through. A cuboid is a prism with a rectangular cross-section; a triangular prism has a triangular one; a cylinder is a prism with a circular cross-section. The single rule for them all is:

Volume of a prism = area of the cross-section × length (or height)

Worked example 5 — triangular prism

A tent shaped like a triangular prism has a triangular cross-section with base 8 cm and height 5 cm, and the prism is 12 cm long.

Step 1 — Area of the triangular cross-section. Area of a triangle = ½ × base × height = ½ × 8 × 5 = 20 cm². Step 2 — Multiply by the length. 20 × 12 = 240 cm³.

Worked example 6 — cylinder

A can has radius 3 cm and height 10 cm. Use π ≈ 3.14.

The cross-section is a circle, so its area = π × r² (the area of a circle).

Step 1 — Area of the circular cross-section: π × 3² = 3.14 × 9 = 28.26 cm². Step 2 — Multiply by the height: 28.26 × 10 = 282.6 cm³.

So volume of a cylinder = π × r² × h. It is exactly the prism rule again, with a circle as the cross-section. To go deeper into circles, see circles: circumference and area.

Watch your units

This is where most marks are lost, so slow down here:

• Length is one dimension → units like cm.
• Area / surface area is two dimensions → square units like cm² (because cm × cm).
• Volume is three dimensions → cubic units like cm³ (because cm × cm × cm).

If your answer comes out in the wrong type of unit, you have almost certainly used the wrong formula. Always ask: am I covering a surface (area) or filling a space (volume)?

Practice activity

Find a real box at home — a cereal box, a tissue box or a shoebox.

1. Use a ruler to measure its length, width and height in centimetres.
2. Calculate its volume using length × width × height.
3. Calculate its surface area by finding the area of all six faces and adding them.
4. Wrap it (in your head) with paper: which number tells you how much paper you need — the surface area or the volume? Write down which, and why.

Then try a cylindrical tin and use the cylinder formula to estimate how much it holds.

Quick recap

Surface area adds up the faces and uses square units; volume fills the inside and uses cubic units. The master formula volume = cross-section × length unlocks cuboids, triangular prisms and cylinders all at once. Keep your units straight and these problems become routine.