IGCSE · Practice questions · Circles, arcs and sectors

Fractions of a circle.

Six original Cambridge-style questions on circles, arcs and sectors: circumference and area, working backwards, exact answers in terms of pi and Extended sector problems.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Find the fraction, then take that much of the circle.

Given: \(C = 2\pi r\) and \(A = \pi r^2\) (List of formulas).
Radius: halve a diameter before using the formulas.
Fraction: sector angle \(\div 360\).
Arc and sector: fraction \(\times 2\pi r\) and fraction \(\times \pi r^2\).
Perimeter of a sector: arc + 2 radii.
Exact: "in terms of \(\pi\)" means keep \(\pi\) as a symbol.

Recall

[2 marks]

A circle has radius 5.2 cm. Calculate (a) its circumference (b) its area.

(a) \(2\pi \times 5.2 = 32.67... = 32.7\) cm ✓
(b) \(\pi \times 5.2^2 = 84.94... = 84.9\text{ cm}^2\) ✓

Calculation

[2 marks]

Without a calculator, find (a) the arc length (b) the area of a sector of radius 6 cm and angle 120°. Give your answers in terms of \(\pi\).

Fraction \(= \dfrac{120}{360} = \dfrac{1}{3}\). ✓
(a) \(\dfrac{1}{3} \times 12\pi = 4\pi\) cm ✓
(b) \(\dfrac{1}{3} \times 36\pi = 12\pi\text{ cm}^2\) ✓

Calculation

[3 marks]

The circumference of a circle is 40 cm. Calculate its area.

\(2\pi r = 40\), so \(r = \dfrac{40}{2\pi} = 6.366...\) cm. ✓
Area \(= \pi \times 6.366...^2 = 127.3...\) ✓
Area \(= 127\text{ cm}^2\) (keep the unrounded radius in the calculator). ✓

Calculation

[3 marks]

(Extended) A major sector of a circle of radius 10 cm has angle 252°. Calculate (a) its area (b) its perimeter.

Fraction \(= \dfrac{252}{360} = 0.7\). ✓
(a) Area \(= 0.7 \times 100\pi = 70\pi = 219.9... = 220\text{ cm}^2\) ✓
(b) Arc \(= 0.7 \times 20\pi = 14\pi = 43.98...\) cm; perimeter \(= 43.98... + 20 = 64.0\) cm ✓

Analysis

[3 marks]

(Extended) A sector of a circle of radius 7 cm has area \(30\text{ cm}^2\).

Calculate (a) the sector angle (b) the arc length.

(a) \(\dfrac{\theta}{360} \times 49\pi = 30\), so \(\theta = \dfrac{30 \times 360}{49\pi} = 70.15... = 70.2\)° ✓
(b) Arc \(= \dfrac{70.15...}{360} \times 14\pi = 8.571... = 8.57\) cm ✓

Analysis

[3 marks]

A bicycle wheel has diameter 66 cm. The bicycle travels 1 km in a straight line.

Calculate the number of complete turns the wheel makes.

One turn moves the bicycle one circumference: \(\pi \times 66 = 207.3...\) cm. ✓
1 km = 100 000 cm. ✓
\(100\,000 \div 207.3... = 482.28...\), so 482 complete turns. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.