Radians

Radians are the standard unit of angle measurement in A-Level Mathematics (9MA0). One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Working in radians simplifies many formulae and is essential for calculus.

Key Definitions

Term	Meaning
Radian	The angle at the centre of a circle when arc length equals radius
Full turn	2pi radians = 360 degrees
Half turn	pi radians = 180 degrees
Quarter turn	pi/2 radians = 90 degrees

Converting Between Degrees and Radians

Degrees to radians: multiply by pi/180

Radians to degrees: multiply by 180/pi

Key Conversions

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

You should memorise these common values.

Worked Example 1

Convert 72 degrees to radians.

72 x pi/180 = 72pi/180 = 2pi/5 radians

Worked Example 2

Convert 5pi/12 radians to degrees.

5pi/12 x 180/pi = 5 x 180/12 = 900/12 = 75 degrees

Arc Length

For a sector with radius r and angle theta (in radians):

Arc length: s = r x theta

Worked Example 3

A sector has radius 8 cm and angle 1.2 radians. Find the arc length.

s = 8 x 1.2 = 9.6 cm

Worked Example 4

An arc of length 15 cm is part of a circle of radius 6 cm. Find the angle in radians.

theta = s/r = 15/6 = 2.5 radians

Sector Area

Area of sector: A = (1/2) x r² x theta

where theta is in radians.

Worked Example 5

Find the area of a sector with radius 10 cm and angle pi/3 radians.

A = (1/2) x 100 x pi/3 = 50pi/3 = 52.36 cm² (to 2 d.p.)

Worked Example 6

A sector has area 24 cm² and radius 6 cm. Find the angle in radians.

24 = (1/2) x 36 x theta

24 = 18 x theta

theta = 24/18 = 4/3 radians

Segment Area

A segment is the region between a chord and the arc.

Area of segment = Area of sector - Area of triangle

= (1/2)r²theta - (1/2)r²sin(theta)

= (1/2)r²(theta - sin(theta))

Worked Example 7

Find the area of a segment of a circle with radius 5 cm and central angle 1.4 radians.

Area = (1/2)(25)(1.4 - sin(1.4))

sin(1.4) = 0.9854...

Area = 12.5 x (1.4 - 0.9854) = 12.5 x 0.4146 = 5.18 cm² (to 2 d.p.)

Exact Trigonometric Values

You must know these exact values (they appear frequently without a calculator):

Angle	sin	cos	tan
0	0	1	0
pi/6 (30)	1/2	sqrt(3)/2	1/sqrt(3) = sqrt(3)/3
pi/4 (45)	sqrt(2)/2	sqrt(2)/2	1
pi/3 (60)	sqrt(3)/2	1/2	sqrt(3)
pi/2 (90)	1	0	undefined

Deriving from Special Triangles

45-45-90 triangle: sides 1, 1, sqrt(2)

sin(pi/4) = 1/sqrt(2), cos(pi/4) = 1/sqrt(2), tan(pi/4) = 1

30-60-90 triangle: sides 1, sqrt(3), 2

sin(pi/6) = 1/2, cos(pi/6) = sqrt(3)/2, tan(pi/6) = 1/sqrt(3)
sin(pi/3) = sqrt(3)/2, cos(pi/3) = 1/2, tan(pi/3) = sqrt(3)

Worked Example 8

Find the exact perimeter of a sector with radius 4 cm and angle pi/3.

Arc length = 4 x pi/3 = 4pi/3

Perimeter = 2 x radius + arc = 8 + 4pi/3

Perimeter = 8 + 4pi/3 cm (exact)

Exam Tips

Tip	Detail
Radian mode	Make sure your calculator is in radian mode for all trig work at A-Level
Leave in exact form	Unless told otherwise, give answers involving pi as exact (e.g. 5pi/6, not 2.618)
Formula links	s = r x theta and A = (1/2)r²theta only work in radians
Segment formula	Remember: segment = sector minus triangle
Memorise exact values	These are tested frequently, especially on Paper 1 (no calculator)

Practice Problems

Convert 150 degrees to radians. [Answer: 5pi/6]
Convert 7pi/4 radians to degrees. [Answer: 315 degrees]
A sector has r = 12 cm and theta = 2pi/3. Find the arc length and area. [Answer: s = 8pi cm, A = 48pi cm²]
Find the area of a segment with r = 8 cm and theta = pi/3. [Answer: (32/3)(pi - 3sqrt(3)/2) or about 5.80 cm²]
A sector has perimeter 20 cm and radius 6 cm. Find the angle. [Answer: 4/3 radians]

Summary

One radian is the angle subtended by an arc of length equal to the radius.
pi radians = 180 degrees.
Arc length: s = r x theta (theta in radians).
Sector area: A = (1/2)r²theta.
Segment area: (1/2)r²(theta - sin theta).
Memorise exact trig values for 0, pi/6, pi/4, pi/3, pi/2.