Rukasha Maths

GCSE maths Beta

Area, circles and volume · Foundation notes

Foundation · Geometry · AQA references G9, G14, G16, G17, G18

Perimeter measures the distance around a shape, area measures its surface, and volume measures the space it occupies. Keep these quantities separate and label them with length, square or cubic units. In area formulae, a height is perpendicular to the chosen base; a sloping side is not automatically the height. For circles, the radius is half the diameter. A prism has the same cross-section all the way through, so its volume is cross-sectional area multiplied by length. Sketch the shape, identify the measurements, and keep π exact until the question asks for a decimal.

Learning objectives

  • Choose area and circumference formulas.
  • Calculate prism volume and give the correct units.

Key ideas

Use a perpendicular height for triangles and parallelograms.

Atriangle=12bh,Aparallelogram=bhA_{\text{triangle}}=\frac12bh,\qquad A_{\text{parallelogram}}=bh

For a trapezium, use the two parallel sides and perpendicular height.

A=12(a+b)hA=\frac12(a+b)h

Distinguish circle area from circumference; radius is half the diameter.

A=πr2,C=2πr=πdA=\pi r^2,\qquad C=2\pi r=\pi d

A prism has a constant cross-section.

V=cross-sectional area×length;Vcylinder=πr2hV=\text{cross-sectional area}\times\text{length};\quad V_{\text{cylinder}}=\pi r^2h

A sector is a fraction of a complete circle.

sector fraction=θ360∘\text{sector fraction}=\frac{\theta}{360^\circ}

Worked examples

Example 1

Find the exact volume of a cylinder with the given radius and height.

r=3 cm,h=8 cm:A=π(3)2=9π cm2r=3\text{ cm},\quad h=8\text{ cm}:\qquad A=\pi(3)^2=9\pi\text{ cm}^2
V=9π×8=72π cm3V=9\pi\times8=72\pi\text{ cm}^3

Answer

72π cm372\pi\text{ cm}^3

Example 2

Find the trapezium area from its parallel sides and perpendicular height.

a=5, b=9, h=4 (cm):A=12(5+9)4=28 cm2a=5,\ b=9,\ h=4\text{ (cm)}:\quad A=\frac12(5+9)4=28\text{ cm}^2

Answer

28 cm228\text{ cm}^2

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Calculate area, circumference, surface area or volume with suitable units.

AO2 · Reasoning

Explain why a length formula cannot be used for an area or volume.

AO3 · Problem solving

Combine shapes or use a volume to determine a missing dimension or capacity.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Use the radius in the circle area formula. Area uses square units; volume uses cubic units.

Try it yourself

Attempt each question before opening its answer.

Check 1: Find the area of a trapezium with parallel sides 5 cm and 9 cm and perpendicular height 4 cm.

Answer: 28 cm²

Area = 1/2 × (5 + 9) × 4 = 28 cm².

Check 2: A circle has diameter 10 cm. Find its area in terms of π.

Answer: 25π cm²

The radius is 5 cm, so area = π × 5² = 25π cm².

Explain it in your own words

  • Explain the units for perimeter, area and volume.
  • Draw a triangle and mark its perpendicular height.
  • For one circle, calculate both area and circumference and explain the different formulae.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes

Specification source

AQA Geometry specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.