Arcs & sectors

Video lesson4 worked examples

Theory

An arc is a fraction of the circumference, and a sector is a fraction of the circle's area.

The formulae rely on the angle at the centre (x°):

\text{Arc Length} = \frac{x}{360} \times \pi d

\text{Sector Area} = \frac{x}{360} \times \pi r^2

You must be able to calculate the length of an arc or the area of a sector, and also work backwards to find the angle at the centre or the radius when given the area/length.

The Golden Rule: everything is the fraction $\frac{x}{360}$ of a whole circle — of the circumference for an arc, of the area for a sector. Write the full formula first, then substitute; to work backwards, substitute what you know and solve for the missing letter.

⚠️ Common Examiner Traps

Radius vs diameter: arc length uses $\pi d$ (diameter); sector area uses $\pi r^2$ (radius). Mixing them up is the classic error.
Major arc / reflex angle: for a major arc, the angle at the centre is the reflex angle $360^\circ - x$ , not the marked one.
Rounding early: keep $\pi$ or full accuracy until the final line.
Which fraction: a working-backwards problem still starts from the same formula — do not invent a new one.

Worked examples

Example 1

Arc Length

Calculate the length of an arc with radius 9 cm and an angle of 40° at the centre.

Step 1: Arc length uses the diameter, so $d = 18$ . Substitute into $\frac{x}{360} \times \pi d$ :

\text{Arc} = \frac{40}{360} \times \pi \times 18

Step 2: Simplify the fraction ( $\frac{40}{360} = \frac{1}{9}$ ) and evaluate: $\frac{1}{9} \times 18\pi = 2\pi$ .

Answer: $2\pi \approx 6.28$ cm.

Example 2

Sector Area

Calculate the area of a sector with a radius of 8 cm and an angle of 45°.

Step 1: Substitute into the area formula: $\text{Area} = \frac{45}{360} \times \pi \times 8^2$ .

Step 2: Simplify the fraction ( $\frac{45}{360} = \frac{1}{8}$ ) and multiply: $\frac{1}{8} \times \pi \times 64$ .

Answer: $8\pi$ cm² (or approx. 25.13 cm²).

Example 3

Working Backwards (Arc Length)

A sector has a radius of 10 cm and an arc length of 15 cm. Find the angle at the centre, x.

Step 1: Set up the arc length equation (note d = 20): $15 = \frac{x}{360} \times \pi \times 20$ .

Step 2: Rearrange to solve for x: $x = \frac{15 \times 360}{20\pi}$ .

Answer: x ≈ 85.9°.

Example 4

🎯 Exam-style (major arc)

A circle has centre O and radius 6 cm. The minor arc AB subtends an angle of 100° at the centre. Calculate the length of the major arc AB.

Step 1: The major arc is the long way round, so its angle is the reflex angle: $360^\circ - 100^\circ = 260^\circ$ .

Step 2: Substitute this angle into the arc-length formula with $d = 12$ :

\text{Major arc} = \frac{260}{360} \times \pi \times 12

Step 3: Evaluate: $\frac{260}{360} \times 12\pi = \frac{26}{3}\pi \approx 27.2$ .

Answer: approximately $27.2$ cm.

Ready to practise?

16 Arcs and Sectors questions with answers and video solutions