A Level Maths 9709 · Paper 1 Pure Mathematics 1 · 1.4 · ~6 min read
Circular measure: radians, arcs and sectors
What a radian really is, converting between degrees and radians, arc length, sector area and segment area, and the multi-step sector problems that combine them with triangles.
The radianDegrees ↔ radiansArc length s = rθSector area ½r²θSegment areaComposite shapesExact answers
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Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end
In 30 seconds
- One radian is the angle at the centre of a circle when the arc length equals the radius. A full turn is 2π radians, so π rad = 180°.
- With θ in radians, arc length s = rθ and sector area A = ½r²θ. These formulas are wrong in degrees.
- Segment = sector − triangle = ½r²(θ − sin θ).
- Most exam questions combine a sector with a triangle: use the sine rule, cosine rule or right-angled trigonometry for the straight edges, and the radian formulas for the curved ones.
By the end you can
- Explain what a radian is and convert between degrees and radians.
- Use s = rθ and A = ½r²θ to find lengths, areas, perimeters and angles.
- Find segment areas and the areas and perimeters of composite shapes.
- Give exact answers in terms of π and surds when asked.
1The radian
Definition and conversions
One radian — the angle subtended at the centre of a circle by an arc equal in length to the radius.
- Why 2π for a full turn? The circumference is 2πr, so it contains 2π arcs of length r. A full turn is 2π radians.
π rad = 180° LEARN IT
degrees → radians: × π/180; radians → degrees: × 180/π| Degrees | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|---|---|
| Radians | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π | 3π/2 | 2π |
- Example: 0.8 rad = 0.8 × 180/π = 45.8°. 1 rad ≈ 57.3°.
Exam tipCheck your calculator mode before every trig calculation in this topic. Radian questions done in degree mode give wrong answers with no method marks for the value.
2Arcs and sectors
The formulas
s = rθ FORMULA LIST (MF19)
arc length, θ in radiansA = ½r²θ FORMULA LIST (MF19)
sector area, θ in radians- Why they work: a sector with angle θ is the fraction θ/(2π) of the whole circle. Arc = (θ/2π) × 2πr = rθ. Area = (θ/2π) × πr² = ½r²θ.
- Perimeter of a sector = r + r + rθ = 2r + rθ. Don't forget the two radii.
Worked example
A sector has radius 8 cm and angle 1.2 radians. Find its arc length, perimeter and area.
- arc s = 8 × 1.2 = 9.6 cm B1
- perimeter = 8 + 8 + 9.6 = 25.6 cm B1
- area = ½ × 8² × 1.2 = 38.4 cm² B1
Worked example
A sector of a circle of radius 6 cm has area 30 cm². Find the angle of the sector.
- ½ × 6² × θ = 30 M1
- θ = 60/36 = 5/3 rad (≈ 1.67 rad) A1
3Segments
Sector minus triangle
segment area = ½r²θ − ½r² sin θ = ½r²(θ − sin θ)
θ in radians; the triangle uses ½ab sin CWorked example
A chord AB of a circle with centre O and radius 6 cm subtends angle π/3 at O. Find the exact area of the minor segment.
- sector = ½ × 36 × π/3 = 6π M1
- triangle = ½ × 36 × sin(π/3) = 18 × √3/2 = 9√3 M1
- segment = 6π − 9√3 cm² (≈ 3.26 cm²) A1
- Chord length: for angle θ at the centre, chord = 2r sin(θ/2). Or use the cosine rule: chord² = r² + r² − 2r² cos θ.
4Composite shapes
Typical exam set-up
- Draw and label the diagram, marking every right angle (a tangent meets a radius at 90°).
- Find straight lengths with trigonometry: SOH CAH TOA in right-angled triangles, the sine or cosine rule otherwise.
- Find curved lengths and areas with s = rθ and ½r²θ.
- Add or subtract regions. Write which ones: "shaded area = triangle OAC − sector OAB".
Worked example
OAB is a sector of a circle with centre O, radius 10 cm and angle AOB = 0.9 rad. The tangent to the arc at A meets OB extended at C. Find the area and perimeter of the region bounded by AC, CB and the arc AB.
- angle OAC = π/2 (tangent ⊥ radius), so AC = 10 tan 0.9 = 12.60 cm and OC = 10/cos 0.9 = 16.09 cm M1 A1
- area = triangle OAC − sector OAB = ½ × 10 × 12.60 − ½ × 10² × 0.9 M1
- = 63.01 − 45 = 18.0 cm² A1
- perimeter = AC + BC + arc AB = 12.60 + (16.09 − 10) + 10 × 0.9 M1
- = 27.7 cm A1
Common mistakeStopping too early. In June 2024, many candidates found a perimeter or an area of one piece and stopped. Re-read the question: "the region bounded by…" usually needs two or three pieces.
Common mistakeRounding in the middle. Keep 12.6015… in your calculator; round only the final answer to 3 significant figures.
Linking questionRadians are used for every calculus result with trig functions in Further differentiation, and for all trig graphs in Trigonometry.
Test yourself — cover the answers first
Quick check
- Convert 225° to radians.
- Convert 2.5 rad to degrees.
- What is the arc length of a sector with radius 4 and angle π/4?
- Write a formula for the area of a segment with angle θ at the centre.
Exam-style
- A sector of radius r has perimeter 20 cm and area 16 cm². Find the possible values of r and θ.
- A chord of a circle of radius 5 cm subtends an angle of 2π/3 at the centre. Find the area of the minor segment and the perimeter of the segment.
Show answers
- 5π/4
- 143.2°
- π
- ½r²(θ − sin θ)
- 2r + rθ = 20 and ½r²θ = 16 ⇒ r² − 10r + 16 = 0 ⇒ r = 2 or 8
r = 2 gives θ = 8 rad, more than a full turn (2π ≈ 6.28), so reject it
r = 8 cm, θ = 0.5 rad - area = 12.5(2π/3 − sin 2π/3) = 15.4 cm²; perimeter = 10π/3 + 5√3 = 19.1 cm
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