Area of a sector and arc length: the formulas in degrees (θ/360 × πr² and θ/360 × 2πr) and in radians (½r²θ and rθ), where they come from, the perimeter of a sector, the area of a segment, working backwards to find the angle or the radius, eight worked examples in terms of π and to 3 significant figures, the mistakes, and a printable formula sheet with practice

"Area of a sector" is searched about 7,400 times a month in the UK, almost entirely by GCSE students, with arc length and the perimeter of a sector asked alongside it on nearly every Foundation and Higher paper, and the same formulas sit in the Australian Year 9 and 10 course, CBSE Class 10 (Areas Related to Circles) and US Geometry. This page gives the two formulas in degrees and radians as fractions of the circle, the perimeter, the segment, the reverse problems, eight worked examples in exact and decimal form, the mistakes, and a printable formula sheet with practice.

Download the sector and arc formula sheet with practice (PDF) (A4, the formulas in both units, the segment and the worked examples on page 1, the reverse problems and the practice set with answers on page 2, free).

The one idea: a fraction of the circle

A sector is the region between two radii and the arc they cut off: a slice of pizza. Its angle at the centre, θ, tells you what fraction of the whole circle it is: θ/360 in degrees, or θ/2π in radians. Everything follows:

The radian forms come from substituting θ/2π for the fraction: θ/2π × πr² = ½r²θ, and θ/2π × 2πr = rθ. Radians are A-level and Australian Year 11; GCSE and Class 10 work in degrees.

Shortcuts: a semicircle (180°) is half: area ½πr², arc πr. A quadrant (90°) is a quarter: area ¼πr², arc ½πr.

Worked examples in degrees

Radius 6 cm, angle 60°:

• Fraction: 60/360 = 1/6.
• Area = 1/6 × π × 6² = 1/6 × 36π = 6π cm² = 18.8 cm² (3 s.f.).
• Arc length = 1/6 × 2π × 6 = 1/6 × 12π = 2π cm = 6.28 cm.
• Perimeter = 2π + 6 + 6 = 2π + 12 = 18.3 cm.

Radius 10 cm, angle 135°:

• Fraction 135/360 = 3/8.
• Area = 3/8 × 100π = 37.5π = 118 cm². Arc = 3/8 × 20π = 7.5π = 23.6 cm. Perimeter = 7.5π + 20 = 43.6 cm.

Radius 4 m, angle 300° (the big sector left when a 60° slice is removed):

• Fraction 300/360 = 5/6. Area = 5/6 × 16π = 13.3π = 41.9 m². Arc = 5/6 × 8π = 20.9 m.

Give answers in terms of π when the question says "exact" or "in terms of π", and to 3 significant figures otherwise; write the exact form first either way, so a rounding slip costs one mark and not all of them.

Worked examples in radians

Radius 5 cm, angle 1.2 radians:

• Area = ½ × 25 × 1.2 = 15 cm². Arc = 5 × 1.2 = 6 cm. Perimeter = 6 + 10 = 16 cm.

Radius 8 cm, angle π/3 (which is 60°):

• Area = ½ × 64 × π/3 = 32π/3 = 33.5 cm². Arc = 8 × π/3 = 8π/3 = 8.38 cm.

Convert when needed: degrees to radians × π/180; radians to degrees × 180/π. 60° = π/3; 1.2 rad = 68.8°.

The segment

A segment is the region between a chord and the arc. Its area is the sector minus the isosceles triangle made by the two radii and the chord:

Segment area = sector area − triangle area = θ/360 × πr² − ½r² sin θ (degrees), or ½r²(θ − sin θ) (radians).

Radius 6 cm, angle 60°: sector 6π = 18.85; triangle ½ × 6 × 6 × sin 60° = 18 × 0.866 = 15.59; segment 3.26 cm². The triangle area uses ½ab sin C, which is why segments are Higher tier and A-level.

The chord length itself is 2r sin(θ/2): for radius 6 and 60°, 2 × 6 × 0.5 = 6 cm (an equilateral triangle, as expected at 60°).

Working backwards

• Angle from area: θ = A ÷ (πr²) × 360. Area 10π, radius 5: θ = 10π ÷ 25π × 360 = 144°.
• Angle from arc: θ = arc ÷ (2πr) × 360. Arc 4π, radius 6: θ = 4π ÷ 12π × 360 = 120°.
• Radius from area and angle: r² = A × 360 ÷ (θπ). Area 24π, angle 60°: r² = 24π × 360 ÷ (60π) = 144, r = 12.
• Radius from arc and angle: r = arc × 360 ÷ (θ × 2π). Arc 5π, angle 150°: r = 5π × 360 ÷ (300π) = 6.
• Radians: θ = 2A ÷ r², θ = arc ÷ r. A sector of radius 4 and arc 10 has θ = 2.5 rad and area ½ × 16 × 2.5 = 20.

Eight worked examples in one table

The mistakes that cost marks

• Using the diameter as r.
• Forgetting the two radii in a perimeter (giving the arc length instead).
• Adding radii to an area, or giving an area in cm.
• Using 2πr for the area fraction or πr² for the arc.
• Mixing units: a radian angle in the degree formula (1.2/360 × πr² is wrong) or 60 in ½r²θ.
• Rounding π early (use the π key, or leave π in until the end).
• Confusing sector and segment, or subtracting the wrong triangle.
• For a "shaded region" question, not stating which areas are being subtracted.

Where it sits in each curriculum

Practice (answers in the PDF)

1. A sector has radius 12 cm and angle 45°. Find its area and arc length in terms of π.
2. Find the perimeter of a sector with radius 7 cm and angle 120°, to 3 s.f.
3. A sector of radius 9 cm has area 27π cm². Find its angle.
4. A sector of angle 72° has arc length 4π cm. Find its radius.
5. (Radians) Find the area and arc length of a sector with radius 10 cm and angle 0.8 rad.
6. (Higher) Find the area of the segment cut off by a chord in a circle of radius 10 cm where the chord subtends 90° at the centre.

Download the formula sheet and the practice set (PDF)

Where to get help

A 1-on-1 online maths class in UK and Australian after-school hours with a tutor who holds a master's in mathematics and teaches GCSE and A-level maths, the Australian Years 9 to 12 and CBSE Class 10 every week: circles taught from the fraction-of-the-circle idea, with every answer written exactly in terms of π before it is rounded. The GCSE maths page explains how the classes run; the free 30-minute first class is a real lesson on this week's homework. Related pages: the trapezium and volumes of a hemisphere, sphere, cone and frustum.

Questions parents ask

See how we teach this, 1-on-1 online →