Volume and Surface Area of a Cylinder: KS3 Maths

A cylinder has a circular cross-section. Its volume is the area of the circular base multiplied by its height: V = πr²h. Its surface area is the sum of the two circular ends and the curved side: SA = 2πr² + 2πrh. Both formulas use the radius r (half the diameter), not the diameter.

What is a cylinder?

A cylinder is a 3-D shape with two identical circular faces (top and bottom) connected by a curved surface. Everyday examples include tins of food, drinking glasses, and candles. In KS3 maths, you need to work out how much space a cylinder occupies (volume) and how much material its surface requires (surface area).

What is the formula for the volume of a cylinder?

Think of a cylinder as a stack of identical circular discs. The area of one disc is πr², and stacking discs of that area to a height h gives:

Volume = πr²h

Quantity | Formula | Units
Volume | πr²h | cm³, m³
Area of circular face | πr² | cm², m²
Circumference of circle | 2πr | cm, m

How do you calculate the volume of a cylinder?

Worked example: A cylinder has radius 4 cm and height 10 cm. Find its volume. Give your answer to 3 significant figures.

Identify values: r = 4 cm, h = 10 cm.

Apply the formula: V = π × 4² × 10.

V = π × 16 × 10 = 160π.

V = 160 × 3.14159… = 502 cm³ (3 s.f.).

Tip: Leave the answer as 160π if the question asks for an exact answer. Only convert to a decimal if told to.

What is the formula for the surface area of a cylinder?

The surface area has three parts:

Two circular ends: each has area πr², so together they contribute 2πr².

Curved surface: if you "unroll" the curved side it forms a rectangle with width equal to the circumference (2πr) and height h, giving area 2πrh.

Surface Area = 2πr² + 2πrh = 2πr(r + h)

How do you calculate the surface area of a cylinder?

Worked example: A cylinder has radius 3 cm and height 8 cm. Find its total surface area. Give your answer to 3 significant figures.

r = 3 cm, h = 8 cm.

Two circular ends: 2 × π × 3² = 2 × π × 9 = 18π.

Curved surface: 2 × π × 3 × 8 = 48π.

Total SA = 18π + 48π = 66π.

SA = 66 × 3.14159… = 207 cm² (3 s.f.).

What if you are given the diameter instead of the radius?

The radius is always half the diameter. If a question states the diameter is 12 cm, r = 12 ÷ 2 = 6 cm. Always halve the diameter before substituting into any formula. Forgetting to halve the diameter is the single most common error in cylinder questions.

How do you find the height if the volume is given?

Rearrange V = πr²h to make h the subject: h = V ÷ (πr²).

Worked example: A cylinder has volume 400 cm³ and radius 5 cm. Find its height.

h = 400 ÷ (π × 5²).

h = 400 ÷ (25π).

h = 400 ÷ 78.5398… = 5.09 cm (3 s.f.).

Frequently asked questions

What units should I use for volume and surface area?

Volume is measured in cubic units (e.g. cm³, m³). Surface area is measured in square units (e.g. cm², m²). If the radius is in centimetres and the height is in centimetres, the volume will automatically be in cm³ — you do not need to convert. However, if the radius is in cm and the height is in mm, convert to the same unit first.

Do I need to memorise the cylinder formulas?

At KS3 most exam papers provide a formulae sheet with area formulas including πr²h for cylinders. However, GCSE papers sometimes ask you to recall formulas, and understanding where they come from (area of circle × height) makes them easier to remember and apply confidently.

What is the difference between the volume of a cylinder and a prism?

Both use the same logic: Volume = cross-sectional area × length (or height). For a cylinder, the cross-section is a circle (area = πr²). For a triangular prism, the cross-section is a triangle. Recognising the pattern means you can find the volume of any prism once you know the area of its cross-section.

How do I use a calculator for π?

Use your calculator's π key (usually labelled π or accessed via SHIFT and a digit) rather than typing 3.14. Using 3.14 introduces rounding errors. Keep π in your calculation until the final step, then round the overall answer to the required accuracy.

For Socratic KS3 geometry and mensuration practice, see aitutors.me.