Circles: circumference, area and parts of a circle (arc, sector, segment)

What you'll learn

This revision guide covers all circle calculations tested at AQA GCSE Mathematics level, including both Foundation and Higher tier content. You'll master the formulas for circumference and area of full circles, then progress to calculating arc lengths, sector areas and segment areas using proportional reasoning and angle relationships. These skills are essential for geometry questions worth 4-6 marks in Paper 2 and Paper 3.

Key terms and definitions

Circumference — the perimeter (distance around the edge) of a circle, calculated using C = πd or C = 2πr

Radius — the distance from the centre of a circle to any point on its circumference, typically denoted by r

Diameter — the distance across a circle through its centre, equal to twice the radius (d = 2r)

Arc — a portion of the circumference of a circle, measured as a curved length between two points

Sector — the region enclosed by two radii and an arc, resembling a "slice of pie"

Segment — the region between a chord and an arc, formed when a chord cuts off part of a circle

Chord — a straight line connecting two points on the circumference of a circle

Major and minor — descriptors for the larger (major) or smaller (minor) version when a circle is divided into two arcs, sectors or segments

Core concepts

Circumference of a circle

The circumference is calculated using either of two equivalent formulas:

C = πd (where d is the diameter)

C = 2πr (where r is the radius)

You must choose the appropriate formula based on the information given in the question. Both formulas give identical results since d = 2r.

Using π in calculations:

• Leave answers in terms of π when instructed (e.g., 12π cm)
• Use the π button on your calculator for numerical answers
• If your calculator lacks a π button, use 3.142 or 3.14159
• Round only the final answer, not intermediate steps

Example calculation: A circle has radius 5 cm. Find its circumference in terms of π.

• C = 2πr = 2π × 5 = 10π cm

For a numerical answer:

• C = 10π = 31.4159... = 31.4 cm (to 1 d.p.)

Area of a circle

The area enclosed by a circle is calculated using:

A = πr²

Note that the radius must be squared before multiplying by π. This is a common error point.

If you're given the diameter instead of the radius, you must halve it first:

• Radius = diameter ÷ 2
• Then apply A = πr²

Example calculation: A circle has diameter 8 cm. Find its area.

• First find the radius: r = 8 ÷ 2 = 4 cm
• Then A = πr² = π × 4² = π × 16 = 16π cm² (in terms of π)
• Or A = 50.3 cm² (to 1 d.p.)

Units: Area is always measured in square units (cm², m², mm²). Circumference uses single units (cm, m, mm).

Arc length

An arc is part of the circumference. To find arc length, use proportional reasoning based on the angle at the centre.

Arc length formula:

Arc length = (θ/360) × πd or Arc length = (θ/360) × 2πr

where θ is the angle at the centre in degrees.

Method:

1. Identify the angle θ at the centre of the circle
2. Calculate what fraction of the full circle this represents (θ/360)
3. Multiply this fraction by the full circumference

Example: Find the arc length AB in a circle of radius 6 cm, where the angle at the centre is 60°.

• Full circumference = 2πr = 2π × 6 = 12π cm
• Fraction of circle = 60/360 = 1/6
• Arc length = 1/6 × 12π = 2π cm
• Or 6.28 cm (to 2 d.p.)

Sector area

A sector is a "slice" of the circle. Its area is calculated using the same proportional approach.

Sector area formula:

Sector area = (θ/360) × πr²

where θ is the angle at the centre in degrees.

Method:

1. Identify the angle θ at the centre
2. Calculate the full circle area (πr²)
3. Multiply by the fraction θ/360

Example: A sector has angle 90° and radius 8 cm. Find its area.

• Full circle area = πr² = π × 8² = 64π cm²
• Fraction = 90/360 = 1/4
• Sector area = 1/4 × 64π = 16π cm²
• Or 50.3 cm² (to 1 d.p.)

Major vs minor sectors: When the angle is greater than 180°, you have a major sector. When less than 180°, it's a minor sector. Ensure you're calculating the correct one based on the diagram.

Segment area

A segment is formed when a chord cuts the circle. Calculating segment area requires finding the sector area and subtracting (or adding) a triangle.

For a minor segment:

Segment area = Sector area - Triangle area

Method:

1. Calculate the sector area using (θ/360) × πr²
2. Calculate the triangle area formed by the two radii and the chord
3. Subtract the triangle area from the sector area

Triangle area in circles: When two radii form an angle θ at the centre, the triangle area is:

• Triangle area = 1/2 × r × r × sin(θ) = (1/2)r²sin(θ)

This formula appears on the AQA formula sheet for Higher tier.

Example: Find the area of the minor segment when a chord subtends an angle of 120° at the centre of a circle with radius 5 cm.

• Sector area = (120/360) × π × 5² = (1/3) × 25π = 25π/3 cm²
• Triangle area = (1/2) × 5² × sin(120°) = 12.5 × 0.866... = 10.825... cm²
• Segment area = 25π/3 - 10.825... = 26.18... - 10.825... = 15.4 cm² (to 1 d.p.)

For a major segment: Major segment area = Sector area + Triangle area (when calculating via the reflex angle), or simply subtract the minor segment from the total circle area.

Perimeter of sectors and segments

The perimeter of a sector includes the arc length plus two radii:

Perimeter of sector = arc length + 2r = (θ/360) × 2πr + 2r

The perimeter of a segment includes the arc length plus the chord length:

Perimeter of segment = arc length + chord length

Calculating the chord length may require trigonometry or Pythagoras' theorem, typically in Higher tier questions.

Worked examples

Example 1: Foundation tier - Circle calculations (4 marks)

Question: A circular pond has a diameter of 3.5 metres. (a) Calculate the circumference of the pond. Give your answer to 1 decimal place. (2 marks) (b) Calculate the area of the pond. Give your answer to 2 decimal places. (2 marks)

Solution:

(a) C = πd C = π × 3.5 ✓ (method mark) C = 11.0 m (to 1 d.p.) ✓ (accuracy mark)

(b) First find radius: r = 3.5 ÷ 2 = 1.75 m A = πr² A = π × 1.75² ✓ (method mark) A = 9.62 m² (to 2 d.p.) ✓ (accuracy mark)

Mark scheme notes: Full marks require correct formula application and accurate rounding. Show working clearly for method marks even if the final answer is incorrect.

Example 2: Higher tier - Arc and sector (5 marks)

Question: The diagram shows a sector OAB of a circle with centre O and radius 9 cm. Angle AOB = 140°.

Calculate: (a) the arc length AB (2 marks) (b) the area of the sector OAB (2 marks) (c) the perimeter of the sector OAB (1 mark)

Give your answers to 3 significant figures.

Solution:

(a) Arc length = (θ/360) × 2πr Arc length = (140/360) × 2π × 9 ✓ Arc length = 22.0 cm (to 3 s.f.) ✓

(b) Sector area = (θ/360) × πr² Sector area = (140/360) × π × 9² ✓ Sector area = 99.0 cm² (to 3 s.f.) ✓

(c) Perimeter = arc length + 2r Perimeter = 22.0... + 18 ✓ Perimeter = 40.0 cm (to 3 s.f.)

Mark scheme notes: The perimeter mark depends on using your arc length from part (a), so don't recalculate unnecessarily. Round only at the final answer stage.

Example 3: Higher tier - Segment area (6 marks)

Question: A chord AB divides a circle of radius 7 cm into two segments. The chord subtends an angle of 80° at the centre O. Calculate the area of the minor segment. Give your answer to 3 significant figures.

Solution:

Step 1: Calculate sector area Sector area = (80/360) × πr² Sector area = (80/360) × π × 7² ✓ Sector area = (2/9) × 49π = 34.2035... cm² ✓

Step 2: Calculate triangle area Triangle area = (1/2) r² sin(θ) ✓ Triangle area = (1/2) × 7² × sin(80°) Triangle area = (1/2) × 49 × 0.98481... ✓ Triangle area = 24.127... cm²

Step 3: Calculate segment area Segment area = Sector area - Triangle area ✓ Segment area = 34.2035... - 24.127... Segment area = 10.1 cm² (to 3 s.f.) ✓

Mark scheme notes: This question requires the trigonometry formula from the formula sheet. Ensure your calculator is in degree mode. Method marks are awarded for correct formula selection even with arithmetic errors.

Common mistakes and how to avoid them

• Confusing radius and diameter — Always check which measurement you're given. If given diameter, halve it before using area formulas. The area formula uses r, not d.

• Forgetting to square the radius — In A = πr², you must calculate r² before multiplying by π. Writing π × r × r helps avoid this error.

• Using 360 instead of the actual angle — When calculating arc length or sector area, substitute the given angle for θ, not 360. The fraction (θ/360) tells you what portion of the circle you have.

• Rounding too early — Keep full calculator values throughout your working. Only round the final answer to the required degree of accuracy. Premature rounding loses marks for accuracy.

• Wrong units in answers — Circumference and arc length use single units (cm, m). Area, sector area and segment area need square units (cm², m²). Check your answer makes sense.

• Calculator in radians mode — For GCSE, all angle measurements are in degrees. Ensure your calculator is in degree mode (DEG) when using sin, cos or tan. Wrong mode gives completely incorrect answers.

Exam technique for circles questions

• Command words matter — "Calculate" requires working shown and a numerical answer. "Find" may accept exact answers in terms of π unless "give your answer to..." specifies otherwise. "Hence" means use your previous answer.

• Use the formula sheet — Higher tier students have access to sector area and segment formulas. Don't waste time memorising these; instead, practice identifying which formula to apply. Foundation students must know C = πd, C = 2πr and A = πr².

• Show your method clearly — Circle questions typically award 1 mark for correct method and 1 mark for accuracy. Even if you make an arithmetic error, clear working secures method marks. Write the formula first, then substitute values.

• Check the degree of accuracy required — Questions specify "to 1 decimal place," "to 3 significant figures," or "in terms of π." Meeting this requirement is worth a mark. If no accuracy is stated, give at least 3 significant figures for a numerical answer.

Quick revision summary

Circles form a key GCSE topic tested across all papers. Master the core formulas: C = πd or 2πr for circumference, A = πr² for area. For parts of circles, use proportional reasoning with angle/360: arc length = (θ/360) × 2πr, sector area = (θ/360) × πr². Segment area requires subtracting triangle area from sector area using the trigonometry formula on the Higher tier formula sheet. Always check whether you're given radius or diameter, ensure correct units, and round only final answers to the specified accuracy.