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Arc length = θ/360 × 2πr, because a sector is the fraction θ/360 of a full circle. Sector area = θ/360 × πr². Free GCSE Maths notes and 11 FAQs.
Arc Length and Sector Area is part of Mensuration and calculation in GCSE AQA Higher Maths.
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Arc length = θ/360 × 2πr, because a sector is the fraction θ/360 of a full circle. Sector area = θ/360 × πr². Free GCSE Maths notes and 11 FAQs.
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Use this lesson to stay inside Mensuration and calculation while connecting the detail back to the wider Geometry and measures topic.
Read the core explanations from Arc Length and Sector Area before testing yourself from memory.
Imagine slicing a pizza. Each slice is a sector — two straight cuts from the centre and a curved crust. The crust length is the arc length; the area of the slice is the sector area. These two quantities appear constantly in Higher-tier problems, often hidden inside multi-step questions about perimeters of composite shapes or areas of shaded regions.
The key insight that unlocks both formulae is this: a sector is a fraction of a full circle. Once you know what fraction, you multiply the full-circle formula by that fraction. That single idea drives everything in this lesson.
A full circle has circumference $2\pi r$ and area $\pi r^2$. A sector with angle $\theta^\circ$ takes up $\frac{\theta}{360}$ of the full circle. Multiply each full-circle formula by that fraction and you have the two sector formulae you need for every question.
Theory is clear, but the method only sticks when you see it in action. Here is a standard exam-style arc length question worked through step by step. Try each step yourself before revealing the answer.
Worked example: Calculating Arc Length
A sector has radius $9$ cm and an angle of $80^\circ$ at the centre. Calculate the arc length. Give your answer in terms of $\pi$.
Answer: $4\pi$ cm (exact) or approximately $12.57$ cm
Now apply the same fraction-of-a-circle logic to area. The structure of the method is identical — the only change is that you multiply by $\pi r^2$ instead of $2\pi r$. Watch how the cancellation works here too.
Worked example: Calculating Sector Area
A sector has radius $12$ cm and an angle of $150^\circ$ at the centre. Calculate the area of the sector. Give your answer to $1$ decimal place.
Answer: $60\pi \approx 188.5$ cm$^2$ (1 d.p.)
GCSE Higher exam questions do not always give you $\theta$ and $r$ and ask for arc length or area. Sometimes the arc length or area is given and you must find the angle or the radius. The method is to substitute what you know into the formula and then rearrange.
This is where the fraction-of-a-circle idea pays off again. If you know the arc length and the radius, you can write $\frac{\theta}{360} = \frac{\text{arc length}}{2\pi r}$ and solve for $\theta$. The same logic applies to area. Let's see this in a worked example.
Worked example: Finding the Angle from Arc Length
A sector has radius $10$ cm and arc length $5\pi$ cm. Calculate the angle $\theta$ at the centre.
Answer: $\theta = 90^\circ$
At Higher tier, arc length and sector area rarely appear in isolation. They are embedded in problems about perimeters of composite shapes, areas of shaded regions, or contexts where you must identify the sector from a diagram. The method stays the same — the challenge is recognising which formula to use and in which order.
A classic exam-style problem asks for the perimeter of a sector. Many students calculate only the arc length and forget the two straight radii. The perimeter of a sector has three parts: the arc and the two radii.
Worked example: Perimeter of a Sector (Multi-Step)
A sector has radius $8$ cm and angle $135^\circ$. Calculate the perimeter of the sector. Give your answer to $3$ significant figures.
Answer: $6\pi + 16 \approx 34.8$ cm (3 s.f.)
You now have the full toolkit. This final section consolidates the mistakes that cost students marks most often, and the habits that win them back. These points are drawn directly from mark schemes patterns.
Every arc length and sector area question on Higher-tier comes back to one idea: the sector is a fraction $\frac{\theta}{360}$ of the full circle. Identify that fraction, multiply by the correct full-circle formula, and handle the algebra carefully — that is the complete method for all three specification requirements: arc lengths, angles, and areas of sectors.
Arc length and sector area questions become easier once you remember that both are fixed fractions of a full circle. The angle at the centre controls the fraction: divide the given angle by 360, then multiply by the full circumference for arc length or the full circle area for sector area. When an exam question mixes radius, diameter and angle information, rewrite the known values first so you do not swap circumference and area formulas. This also helps when comparing major and minor sectors, because you can see immediately which fraction of the circle each region represents.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Arc length = θ/360 × 2πr, because a sector is the fraction θ/360 of a full circle. For radius 9 cm and an angle of 80°, the arc length is 80/360 × 2π × 9 = 2/9 × 18π = 4π cm.
Sector area = θ/360 × πr². For radius 12 cm and an angle of 150°, the area is 150/360 × π × 144 = 5/12 × 144π = 60π ≈ 188.5 cm².
Add the arc length and both radii. For radius 8 cm and an angle of 135°, the arc is 135/360 × 2π × 8 = 6π cm, so the perimeter is 6π + 16 ≈ 34.8 cm to 3 s.f.
A sector is a slice of a circle made by two straight cuts from the centre and a curved arc, like a slice of pizza. A sector with angle θ° is θ/360 of the full circle, which is why both sector formulae multiply by that fraction.
An arc is part of the circumference of a circle, like the curved crust of a pizza slice. In a sector, the arc length is θ/360 × 2πr, where θ is the angle at the centre.
A major sector is the larger part of a circle and a minor sector is the slice left over, and together they make the full circle. Dividing each angle by 360 shows what fraction of the circle each region represents.
Substitute into the arc length formula and solve for θ. For radius 10 cm and arc length 5π cm, 5π = θ/360 × 20π, so θ/360 = 1/4 and θ = 90°.
A sector with radius 9 cm and angle 80° has arc length 4π cm, about 12.57 cm. Using arc length = θ/360 × 2πr, 80/360 = 2/9, and 2/9 × 18π = 4π.
A sector with radius 12 cm and angle 150° has area 60π cm², about 188.5 cm². Using area = θ/360 × πr², 150/360 = 5/12 and 12² = 144, so 5/12 × 144π = 60π.
A sector with angle θ° at the centre is θ/360 of the full circle. Its arc length is that fraction of 2πr and its area is that fraction of πr², so a 90° sector is a quarter circle.
The radius is found from the arc length by substituting the arc length and angle into arc length = θ/360 × 2πr, then rearranging to solve for r. The same method works with the sector area formula.