Arc length and sector area calculator

Radius 6 cm, angle 60°. The sector is $60 \div 360 = \tfrac{1}{6}$ of the circle.

• Arc length $= \tfrac{1}{6} \times 2\pi \times 6 = 2\pi \approx 6.28$ cm.
• Sector area $= \tfrac{1}{6} \times \pi \times 36 = 6\pi \approx 18.85$ cm².
• Perimeter $= 2\pi + 6 + 6 \approx 18.28$ cm.

The perimeter is the arc plus the two straight radii. Forgetting the radii is the most common mistake in these questions: the arc alone is just the curved edge.

• Finding the angle from an arc length: $\theta = \dfrac{\text{arc}}{2\pi r} \times 360$. An arc of 5 cm on a circle of radius 10 cm gives $\theta = 5 \div 62.83 \times 360 \approx 28.6°$.
• Finding the radius from the sector area: $r = \sqrt{\dfrac{\text{area} \times 360}{\theta \pi}}$.

Rearranging these is a common Higher-tier exam question.

A segment is the region between a chord and the arc. Its area is the sector area minus the area of the triangle formed by the two radii and the chord: $\tfrac{1}{2} r^2 \sin\theta$. For radius 6 and 60°, the triangle is $\tfrac{1}{2} \times 36 \times \sin 60° \approx 15.59$, so the segment is $18.85 - 15.59 \approx 3.26$ cm².

• Using 180 instead of 360 in the fraction.
• Using the diameter as the radius.
• Leaving the two radii out of the perimeter.

Practise with our arc length and sector area questions, and use the circle calculator for whole circles.

• Write the fraction $\tfrac{\theta}{360}$ first: it shows the method even if a later step slips.
• Exact answers keep $\pi$: $2\pi$ cm, $6\pi$ cm².
• Radians appear at A level; at GCSE and IGCSE, angles are in degrees.

• Pie charts: each slice is a sector. A category that's 25% of the data gets $0.25 \times 360 = 90°$, and its share of the pie's area is also 25%.
• Clock hands: the minute hand of a clock sweeps 6° each minute. In 20 minutes it turns 120°, so the tip of a 10 cm hand travels an arc of $\tfrac{120}{360} \times 2\pi \times 10 \approx 20.9$ cm.
• Windscreen wipers and sprinklers cover sectors (or the region between two arcs), which is why these shapes appear so often in exam questions.

When a question gives the arc length or area and asks for the angle or radius, rearrange the same formulas: the calculator's working shows the fraction $\tfrac{\theta}{360}$ you need.