Area, Surface Area and Volume
Area measures a flat surface, surface area measures the outside of a three-dimensional object, and volume measures the space inside it. Learn how to choose the right formula, show your working and use correct square or cubic units.
Estimated learning time: 30–40 minutes · Beginner to intermediate
Learning goals
- Calculate the area of common two-dimensional shapes.
- Find the surface area of cubes, cuboids and simple prisms.
- Calculate the volume of cubes, cuboids and prisms.
- Convert units and solve practical measurement problems.
1. Area, surface area and volume: what is the difference?
| Measurement | What it measures | Typical units |
|---|---|---|
| Area | The amount of flat surface inside a 2D shape | cm², m², km² |
| Surface area | The total area of all outside faces of a 3D object | cm², m² |
| Volume | The space occupied or capacity inside a 3D object | cm³, m³ |
Imagine covering a floor with tiles: you need area. Painting every outside face of a box requires surface area. Filling the box with sand requires volume.
2. Area of common 2D shapes
Use the perpendicular height, not a sloping side, when a formula requires height.
| Shape | Area formula | What the letters mean |
|---|---|---|
| Rectangle | length × width | l is length; w is width |
| Square | side × side = s² | s is one side |
| Triangle | ½ × base × perpendicular height | b is base; h is perpendicular height |
| Parallelogram | base × perpendicular height | h meets the base at 90° |
| Trapezium (trapezoid) | ½ × (a + b) × h | a and b are parallel sides |
| Circle | π × r² | r is radius; π ≈ 3.14 |
A rectangular garden is 8 m long and 5 m wide.
Area = 8 × 5 = 40 m²
The garden covers 40 square metres.
A triangle has base 10 cm and perpendicular height 6 cm.
Area = ½ × 10 × 6 = 30 cm²
The base is 9 cm and perpendicular height is 4 cm.
Area = 9 × 4 = 36 cm²
Do not multiply the base by the sloping side.
The parallel sides are 7 cm and 11 cm. The perpendicular height is 5 cm.
Area = ½ × (7 + 11) × 5 = 45 cm²
A circle has radius 3 cm.
Area = π × 3² = 9π ≈ 28.27 cm²
Radius is half the diameter. If the diameter is given, divide it by 2 first.
The triangle's height meets its base at a right angle.
3. Composite shapes
A composite shape is made from two or more familiar shapes. Split it into simpler parts, calculate each area, and then add or subtract as the diagram requires.
Imagine a 10 m by 8 m rectangle with a 4 m by 3 m rectangle removed from one corner.
Large rectangle area = 10 × 8 = 80 m².
Removed rectangle area = 4 × 3 = 12 m².
L-shape area = 80 − 12 = 68 m²
4. Surface area of 3D objects
Surface area is the sum of the areas of every outside face. A net is a flat layout of those faces that can fold into a 3D object.
A cube has six equal square faces. If each edge is 4 cm, one face has area 4 × 4 = 16 cm².
Surface area = 6 × 16 = 96 cm²
Formula: 6s².
A box is 5 cm long, 3 cm wide and 2 cm high. Its faces come in three matching pairs.
Top and bottom: 2 × (5 × 3) = 30 cm².
Front and back: 2 × (5 × 2) = 20 cm².
Left and right: 2 × (3 × 2) = 12 cm².
Total surface area = 30 + 20 + 12 = 62 cm²
Formula: 2(lw + lh + wh).
A prism has a triangular cross-section with area 12 cm² and perimeter 12 cm. The prism is 8 cm long.
Two triangular ends: 2 × 12 = 24 cm².
Three rectangular side faces together: perimeter × length = 12 × 8 = 96 cm².
Surface area = 24 + 96 = 120 cm²
5. Volume of cubes, cuboids and prisms
Volume tells you how many cubic units fit inside a solid. For a prism, the cross-section stays the same along its length.
| Solid | Volume formula |
|---|---|
| Cube | side × side × side = s³ |
| Cuboid | length × width × height |
| Any prism | area of cross-section × length |
| Cylinder | π × radius² × height |
A storage box is 8 cm long, 3 cm wide and 4 cm high.
Volume = 8 × 3 × 4 = 96 cm³
The triangular cross-section has base 6 cm and perpendicular height 4 cm. The prism is 10 cm long.
Triangle area = ½ × 6 × 4 = 12 cm².
Volume = 12 × 10 = 120 cm³
A cylinder has radius 2 cm and height 7 cm.
Volume = π × 2² × 7 = 28π ≈ 87.96 cm³
6. Units and conversions
Convert measurements before substituting into a formula. For area and volume, the conversion factor must also be squared or cubed.
| Length | Area | Volume |
|---|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² | 1 m³ = 1,000,000 cm³ |
| 1 cm = 10 mm | 1 cm² = 100 mm² | 1 cm³ = 1,000 mm³ |
Convert 2 m² into cm². Since 1 m = 100 cm, then 1 m² = 100 × 100 = 10,000 cm².
2 m² = 2 × 10,000 = 20,000 cm².
Convert 0.5 m³ into cm³. Since 1 m = 100 cm, then 1 m³ = 100 × 100 × 100 = 1,000,000 cm³.
0.5 m³ = 500,000 cm³.
7. Choose the right measurement
- How much carpet covers a room? Find area.
- How much paint covers the outside of a box? Find surface area, then consider paint coverage if required.
- How much water fits inside a tank? Find volume or capacity.
For water, 1 cm³ = 1 mL and 1,000 cm³ = 1 L. Also, 1 m³ = 1,000 L. These links are useful when a volume question asks for liquid capacity.
A rectangular tank measures 50 cm × 30 cm × 40 cm.
Volume = 50 × 30 × 40 = 60,000 cm³.
Since 1,000 cm³ = 1 L, capacity = 60 L.
8. Common mistakes to avoid
- Using a sloping side instead of perpendicular height in a triangle or parallelogram.
- Forgetting the ½ in the triangle and trapezium formulas.
- Finding only one face of a solid when the question asks for total surface area.
- Using a square unit for volume or a cubic unit for area.
- Mixing centimetres and metres in the same calculation without converting.
- Multiplying an area conversion by 100 instead of 100².
- Confusing radius with diameter in a circle formula.
9. Practice questions
Attempt each problem before revealing the worked answer.
- A rectangle is 12 cm long and 7 cm wide. Find its area.
Show solution
12 × 7 = 84 cm².
- A triangle has base 9 m and perpendicular height 4 m. Find its area.
Show solution
½ × 9 × 4 = 18 m².
- A trapezium has parallel sides 6 cm and 10 cm, with height 5 cm. Find its area.
Show solution
½ × (6 + 10) × 5 = 40 cm².
- A cube has edges of 3 cm. Find its total surface area.
Show solution
6 × 3² = 6 × 9 = 54 cm².
- A cuboid measures 6 cm × 4 cm × 2 cm. Find its total surface area.
Show solution
2(6×4 + 6×2 + 4×2) = 2(24 + 12 + 8) = 88 cm².
- A box measures 9 cm × 5 cm × 4 cm. Find its volume.
Show solution
9 × 5 × 4 = 180 cm³.
- A triangular prism has cross-sectional area 15 cm² and length 8 cm. Find its volume.
Show solution
15 × 8 = 120 cm³.
- Convert 3 m² into cm².
Show solution
1 m² = 10,000 cm², so 3 m² = 30,000 cm².
10. Quick knowledge check
Choose one answer for each question, then select Check answers to see your score and explanations.
0 of 5 answered
11. Printable worksheet
Show your calculations and include the correct units.
Area, surface area and volume — Grade 7
Name: __________________________ Date: ______________
- Find the area of a rectangle measuring 15 cm by 6 cm.
- Find the area of a triangle with base 12 cm and perpendicular height 7 cm.
- A parallelogram has base 11 m and perpendicular height 4 m. Find its area.
- A circle has radius 5 cm. Find its area in terms of π and approximately to 2 decimal places.
- Find the total surface area of a cube with edge 5 cm.
- Find the total surface area of a cuboid measuring 8 cm × 3 cm × 2 cm.
- Find the volume of a cuboid measuring 7 m × 4 m × 2 m.
- A prism has cross-sectional area 18 cm² and length 9 cm. Find its volume.
- Convert 4 m² into cm².
- A tank measures 40 cm × 25 cm × 20 cm. Find its capacity in litres.
Answer key
- 15 × 6 = 90 cm².
- ½ × 12 × 7 = 42 cm².
- 11 × 4 = 44 m².
- 25π cm² ≈ 78.54 cm².
- 6 × 5² = 150 cm².
- 2(8×3 + 8×2 + 3×2) = 2(24 + 16 + 6) = 92 cm².
- 7 × 4 × 2 = 56 m³.
- 18 × 9 = 162 cm³.
- 4 × 10,000 = 40,000 cm².
- 40 × 25 × 20 = 20,000 cm³ = 20 L.