A radian is the angle at the centre of a circle when the arc length equals the radius, so a full turn is 2π radians and π radians = 180°. In radians, arc length is s = rθ and sector area is A = ½r²θ. These two formulae, used forwards and backwards, handle sectors, segments and the composite shapes that exam questions are built from.
Radian measure formulae
- π radians = 180°, and 1 radian ≈ 57.3°
- Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π
- Arc length: s = rθ
- Sector area: A = ½r²θ
- Segment area: ½r²θ − ½r² sin θ
- Sector perimeter: rθ + 2r
Every formula here needs θ in radians.
What is a radian?
One radian is the angle at the centre of a circle subtended by an arc whose length equals the radius. A whole circumference is 2πr, so a full turn is 2π radians, and π radians = 180°.
The special angles become 30° = π/6, 45° = π/4, 60° = π/3 and 90° = π/2. Leave radian answers as exact multiples of π wherever you can.
Arc length and sector area
For a sector of radius r and angle θ in radians, the arc length is s = rθ and the area is A = ½r²θ. If the angle is given in degrees, convert it first.
The perimeter of a sector is the arc plus two radii, rθ + 2r. Forgetting the two radii is the single most common dropped mark in this topic.
Finding r or θ from an arc or an area
Many questions give you the arc length or the area and ask for the radius or the angle. Rearrange: θ = s / r, θ = 2A / r², or r = √(2A / θ). These often lead to surds, and if the question says “exact”, keep everything as surds and multiples of π rather than reaching for decimals.
How do you find the area of a segment?
A segment is the region between a chord and its arc. Take the triangle away from the sector: segment area = ½r²θ − ½r² sin θ. Both terms use θ in radians, so sin θ must be worked out with the calculator in radian mode.
The chord length comes from the cosine rule in the isosceles triangle with two sides r and included angle θ. A segment's perimeter is the arc plus the chord.
Composite shapes and applied problems
Exam diagrams combine sectors with triangles: logos, patios, cross-sections. Split the shape into simple pieces, find each area or length separately, then combine. A shaded region is usually a difference, such as a triangle minus a sector, or a sum.
For a perimeter, trace the boundary and include only the edges you actually walk along. For every arc, identify its centre and radius. A radius drawn inside a shape is a construction line, not part of the perimeter. Carry full accuracy until the final answer.
Worked examples
Example 1. (i) Convert 150° to radians, exactly. (ii) Convert 2.8 radians to degrees, to 1 d.p.
- (i) 150 × π/180 = 5π/6.
- (ii) 2.8 × 180/π ≈ 160.4°.
Answer: 5π/6; 160.4°.
Marks: B1 for the exact radians, B1 for the degrees.
Example 2. A sector has radius 7 cm and angle 0.8 radians. Find (i) the arc length and (ii) the area.
- (i) s = rθ = 7 × 0.8 = 5.6 cm.
- (ii) A = ½r²θ = ½ × 49 × 0.8 = 19.6 cm².
Answer: Arc 5.6 cm, area 19.6 cm².
Marks: M1 for the formulae, A1 for both.
Example 3. A sector has angle 1.25 radians and area 15 cm². Find the exact radius and the exact perimeter.
- ½r²(1.25) = 15, so r² = 30 / 1.25 = 24.
- r = √24 = 2√6 cm.
- Perimeter = rθ + 2r = 2√6 × 1.25 + 4√6 = 2.5√6 + 4√6 = 13√6 / 2 cm.
Answer: r = 2√6 cm; perimeter 13√6 / 2 cm.
Marks: M1 for forming the equation, A1 for 2√6, A1 for the perimeter.
Example 4. A sector has radius 6 cm and angle 1.2 radians. Find the area of the segment cut off by the chord, to 2 d.p.
- Sector: ½ × 36 × 1.2 = 21.6 cm².
- Triangle: ½ × 6 × 6 × sin 1.2 ≈ 16.777 cm², in radian mode.
- Segment: 21.6 − 16.777 ≈ 4.82 cm².
Answer: 4.82 cm².
Marks: M1 for the sector, M1 for the triangle, A1 for 4.82.
Example 5. In triangle ABC, AB = 8 cm, AC = 11 cm and angle BAC = 0.7 radians. Arc BD has centre A and radius 8 cm, with D on AC. Find the area of the region bounded by BC, CD and the arc BD, to 2 d.p.
- Triangle ABC: ½ × 8 × 11 × sin 0.7 ≈ 28.346 cm².
- Sector ABD: ½ × 64 × 0.7 = 22.4 cm².
- Region: 28.346 − 22.4 ≈ 5.95 cm².
Answer: 5.95 cm².
Marks: M1 for the triangle, M1 for the sector, A1 for 5.95.
Common mistakes
- Using degrees in s = rθ or ½r²θ. Both formulae only work with θ in radians. Convert first.
- Leaving out the two radii. A sector's perimeter is the arc plus 2r, not just the arc.
- Calculator in degree mode for sin θ. In the segment formula, sin 1.2 means sin of 1.2 radians.
- Rounding part-way through. Keep full accuracy for each piece and round only the final answer.
- Counting a construction line in the perimeter. Trace the boundary of the shape and include only the edges on it.
How radians, arcs and sectors are examined
In Edexcel International A-Level, radians are Pure 1 content (paper WMA11), with a chapter of their own. In Edexcel A-Level (9MA0), radian measure is Year 2 Pure. Typical questions:
- convert between degrees and radians
- find an arc length, sector area or perimeter, often exactly in surd form
- find r or θ from a given area or arc length
- find the area of a segment, or of a composite region in a diagram
The Pure 1 radians exam questions include full diagram-based questions with mark schemes. The radians worksheet and the trigonometry revision engine give extra practice.
Composite figures often need the sine and cosine rules as well, and radians are used throughout calculus with trig functions.
Frequently asked questions
Why do we use radians?
Radians make circle formulae simple: s = rθ and A = ½r²θ. They are also essential in calculus, where results such as the derivative of sin x being cos x only hold in radians.
How many degrees is 1 radian?
About 57.3°, because π radians = 180° and 180 / π ≈ 57.3.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, like a slice of pizza. A segment is bounded by a chord and an arc, which is the sector with the triangle removed.