Radians, Arc Length & Sector Area

This lesson covers radian measure, the formulae for arc length and sector area, and calculations involving segments of circles. Radian measure is essential throughout A-Level Mathematics — it simplifies many formulae in calculus, trigonometry, and mechanics, and is used exclusively in the AQA specification from this point onward.

What Is a Radian?

A radian is a unit for measuring angles. One radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

Since the circumference of a circle is 2πr, a full turn (360°) corresponds to an arc length of 2πr, which gives:

2π radians = 360°

Therefore:

π radians = 180°

This single relationship is the key to all conversions between degrees and radians.

Converting Between Degrees and Radians

Degrees to Radians

Multiply the angle in degrees by π/180:

θ (radians) = θ (degrees) × π/180

Example 1: Convert 45° to radians.

45° × π/180 = π/4 radians

Example 2: Convert 120° to radians.

120° × π/180 = 2π/3 radians

Radians to Degrees

Multiply the angle in radians by 180/π:

θ (degrees) = θ (radians) × 180/π

Example 3: Convert 5π/6 radians to degrees.

5π/6 × 180/π = 150°

Example 4: Convert 1.2 radians to degrees.

1.2 × 180/π = 68.8° (3 s.f.)

Key Radian–Degree Equivalences

Degrees	Radians
0°	0
30°	π/6
45°	π/4
60°	π/3
90°	π/2
120°	2π/3
135°	3π/4
150°	5π/6
180°	π
270°	3π/2
360°	2π

You must know these by heart. In AQA examinations, angles will frequently be given in radians, and you are expected to be fluent in working with them.

Arc Length

An arc is a section of the circumference of a circle. The length of an arc is proportional to the angle it subtends at the centre.

Formula (Radians)

If the radius is r and the angle at the centre is θ (in radians):

s = rθ

where s is the arc length.

This formula is strikingly simple — and that simplicity is one of the main reasons we use radians in advanced mathematics.

Derivation

The full circumference is 2πr, corresponding to a full angle of 2π radians. The fraction of the circle used is θ/(2π). Therefore:

s = 2πr × θ/(2π) = rθ

Example 5: A sector has radius 8 cm and angle 1.5 radians. Find the arc length.

s = rθ = 8 × 1.5 = 12 cm

Example 6: A sector has radius 6 cm and arc length 4π cm. Find the angle in radians.

s = rθ
4π = 6θ
θ = 4π/6 = 2π/3 radians

Example 7: The arc length of a sector is 15 cm and the angle is 2.5 radians. Find the radius.

s = rθ
15 = r × 2.5
r = 15/2.5 = 6 cm

Formula (Degrees)

If the angle is given in degrees, the arc length formula is:

s = (θ/360) × 2πr = πrθ/180

However, at A-Level you should almost always work in radians.

Sector Area

A sector is the region enclosed by two radii and the arc between them (think of a slice of pizza).

Formula (Radians)

A = ½r²θ

where r is the radius and θ is the angle in radians.

Derivation

The area of the full circle is πr², and the fraction used is θ/(2π):

A = πr² × θ/(2π) = ½r²θ

Example 8: Find the area of a sector with radius 10 cm and angle π/4 radians.

A = ½r²θ = ½ × 10² × π/4 = ½ × 100 × π/4 = 25π/2 cm²

This is approximately 39.3 cm².

Example 9: A sector has area 27 cm² and radius 6 cm. Find the angle in radians.

A = ½r²θ
27 = ½ × 36 × θ
27 = 18θ
θ = 27/18 = 3/2 = 1.5 radians

Example 10: A sector has area 50 cm² and angle 0.8 radians. Find the radius.

A = ½r²θ
50 = ½ × r² × 0.8
50 = 0.4r²
r² = 125
r = √125 = 5√5 cm ≈ 11.2 cm

Perimeter of a Sector

The perimeter of a sector consists of two radii and the arc:

Perimeter = 2r + rθ = r(2 + θ)

Example 11: Find the perimeter of a sector with radius 7 cm and angle 1.2 radians.

Perimeter = 7(2 + 1.2) = 7 × 3.2 = 22.4 cm

Area of a Segment

A segment is the region between a chord and the arc it cuts off. There are two types:

Minor segment: the smaller region.
Major segment: the larger region.

Formula

The area of a segment is found by subtracting the area of the triangle from the area of the sector:

Area of segment = Area of sector − Area of triangle
                = ½r²θ − ½r²sinθ
                = ½r²(θ − sinθ)

This uses the triangle area formula ½r²sinθ (since the triangle has two sides of length r and the included angle θ).

Example 12: Find the area of the minor segment of a circle with radius 12 cm and central angle π/3 radians.

Area = ½r²(θ − sinθ)
     = ½ × 144 × (π/3 − sin(π/3))
     = 72 × (π/3 − √3/2)
     = 72π/3 − 72√3/2
     = 24π − 36√3
     ≈ 75.4 − 62.4
     = 13.0 cm² (3 s.f.)

Example 13: A chord AB subtends an angle of 2 radians at the centre of a circle with radius 5 cm. Find the area of the minor segment.

Area = ½ × 25 × (2 − sin 2)
     = 12.5 × (2 − 0.9093...)
     = 12.5 × 1.0907...
     = 13.6 cm² (3 s.f.)

Worked Exam-Style Problem

Problem: The diagram shows a sector OAB of a circle, centre O, radius 8 cm. The angle AOB is 0.9 radians. Find:

(a) the length of arc AB

(b) the area of sector OAB

Solution:

(a) Arc length:

s = rθ = 8 × 0.9 = 7.2 cm

(b) Sector area:

A = ½r²θ = ½ × 64 × 0.9 = 28.8 cm²

A = ½r²(θ − sinθ) = ½ × 64 × (0.9 − sin 0.9)
  = 32 × (0.9 − 0.7833...)
  = 32 × 0.1167...
  = 3.73 cm² (3 s.f.)

Summary

1 radian is the angle at the centre of a circle subtended by an arc equal in length to the radius.
π radians = 180° — use this to convert between degrees and radians.
Arc length: s = rθ (θ in radians).
Sector area: A = ½r²θ (θ in radians).
Sector perimeter: P = r(2 + θ).
Segment area: A = ½r²(θ − sinθ).
Always check that your angle is in radians before using these formulae.

Exam Tip: The formulae s = rθ and A = ½r²θ are in the AQA formula booklet, but you should know them by heart. Remember that these formulae only work when θ is in radians — if the question gives an angle in degrees, convert it first. Always include units in your final answer and give answers in exact form (involving π or surds) where appropriate.