Area, surface area and volume

Find areas of 2D shapes and volumes and surface areas of prisms, cylinders, cones and spheres, including scaling.

Also known as: perimeter, volume of a cylinder, volume of a cone, scale factor.

SAT · Geometry and Trigonometry GCSE Maths · Geometry and measures IGCSE Maths · Mensuration

Key ideas

Triangle $\frac{1}{2}bh$ , trapezium $\frac{1}{2}(a + b)h$ , circle $\pi r^2$ .
Prism or cylinder volume = cross-sectional area × length; cylinder $\pi r^2 h$ .
Cone $\frac{1}{3}\pi r^2 h$ ; sphere $\frac{4}{3}\pi r^3$ ; pyramid $\frac{1}{3}$ × base × height.
If lengths scale by $k$ , areas scale by $k^2$ and volumes by $k^3$ .

Area

Split compound shapes into rectangles, triangles and circle parts. A rectangle $8 \times 5$ with a semicircle of diameter $8$ on one side has area $40 + \frac{1}{2}\pi(4)^2 = 40 + 8\pi$ .

Volume and surface area

A cylinder with radius $3$ and height $10$ has volume $\pi(3)^2(10) = 90\pi$ . Its surface area is two circles plus the curved surface: $2\pi r^2 + 2\pi r h = 18\pi + 60\pi = 78\pi$ .

Scale factors

Doubling every length of a cube multiplies its surface area by $2^2 = 4$ and its volume by $2^3 = 8$ . Use this to compare similar shapes without recalculating.

Worked examples

Example 1 · Cylinder volume

A cylindrical tank has radius 3 m and height 10 m. Find its volume in terms of π.

Step 1. $V = \pi r^2 h = \pi \times 9 \times 10$ .

Answer: $90\pi$ m³

Example 2 · Similar solids

Two similar boxes have heights 4 cm and 6 cm. The smaller has volume 64 cm³. Find the volume of the larger.

Step 1.Length scale factor $k = \frac{6}{4} = 1.5$ .
Step 2.Volume factor $k^3 = 3.375$ , so $64 \times 3.375 = 216$ .

Answer: 216 cm³

Common mistakes

Using diameter instead of radius in circle and cylinder formulas.
Scaling volume by k instead of k³ for similar solids.
Mixing units (cm and m) in the same calculation.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Area, surface area and volumeEasier

A rectangular box measures 4 cm by 5 cm by 6 cm. What is its volume, in cubic centimetres?

Question 2Area, surface area and volumeEasier

What is the area of a triangle with base 10 and height 7?