Radian Measure, Arc Length, and Sector Area — Edexcel A-level Mathematics (Pure)

SPEC 5.1 • TRIGONOMETRY

Radian Measure, Arc Length & Sector Area

Converting between degrees and radians, and calculating arc lengths, sector areas, and segment areas

Learning Objectives

By the end of this lesson you will be able to:

Understand the definition of a radian and why radians are used

Convert between degrees and radians

Calculate arc length using

s = r θ

Calculate sector area using

A = \frac{1}{2} r^{2} θ

Calculate the area of a segment

Solve problems involving arcs, sectors, and segments

Exam Note

These formulas are in the formula booklet, but you must use radians (not degrees) for them to work!

Introduction: Why Radians?

Degrees are an arbitrary unit – 360° in a circle is based on ancient Babylonian mathematics (base 60 system).

Radians are a natural unit based on the geometry of the circle itself:

Key Insight

When we use radians, the formulas for arc length and sector area become beautifully simple – no need for awkward fractions or π conversions!

With Degrees

Arc length = \frac{θ}{360°} \times 2 π r

Complicated!

With Radians

Arc length = r θ

Simple!

Definition of a Radian

Definition

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

Key Relationship

The circumference of a circle =

2 π r

So a full turn =

2 π

radians

$360° = 2 π$ radians

$180° = π$ radians

1 radian ≈ 57.3°

Converting Degrees to Radians

Since

180° = π

radians, we can derive the conversion:

Radians = Degrees \times \frac{π}{180}

Method

To convert degrees to radians: multiply by $\frac{π}{180}$

Examples:

90° = 90 \times \frac{π}{180} = \frac{π}{2}

60° = 60 \times \frac{π}{180} = \frac{π}{3}

45° = 45 \times \frac{π}{180} = \frac{π}{4}

Converting Radians to Degrees

The reverse conversion:

Degrees = Radians \times \frac{180}{π}

Method

To convert radians to degrees: multiply by $\frac{180}{π}$

Examples:

\frac{π}{6} = \frac{π}{6} \times \frac{180}{π} = 30°

\frac{2 π}{3} = \frac{2 π}{3} \times \frac{180}{π} = 120°

1.5 rad = 1.5 \times \frac{180}{π} = 85.9°

Calculator Tip

Most calculators have a DRG button to switch between Degrees, Radians, and Gradians. Check your mode before calculating!

Degree-Radian Conversions

Worked Example01

(a) Convert 135° to radians, giving your answer in terms of π.

(b) Convert

\frac{5 π}{6}

radians to degrees.

Part (a)

135° = 135 \times \frac{π}{180} = \frac{135 π}{180} = \frac{3 π}{4}

Part (b)

\frac{5 π}{6} = \frac{5 π}{6} \times \frac{180}{π} = \frac{5 \times 180}{6} = \frac{900}{6} = 150°

Key Radian Values

You should memorise these common conversions:

Degrees0°30°45°60°90°180°270°360°

Radians0

\frac{π}{6}

\frac{π}{4}

\frac{π}{3}

\frac{π}{2}

π

\frac{3 π}{2}

2 π

Memory Tip

For 30°, 45°, 60°: the denominators are 6, 4, 3 (decreasing). The angles 30°, 45°, 60° go up, so the fractions

\frac{π}{6}, \frac{π}{4}, \frac{π}{3}

also increase!

Arc Length Formula

Formula (in booklet)

l = r θ

where

l

= arc length,

r

= radius,

θ

= angle in radians

Understanding the Formula

Arc length is a fraction of the circumference:

l = \frac{θ}{2 π} \times 2 π r = r θ

The

2 π

cancels out when θ is in radians!

Critical

The angle must be in radians for this formula to work. Always convert degrees to radians first!

Finding Arc Length

Worked Example02

A sector of a circle has radius 8 cm and angle 0.75 radians. Find the arc length.

Step 1 Identify values:

r = 8

,

θ = 0.75

rad

Step 2 Apply formula:

l = r θ

l = 8 \times 0.75 = 6 cm

Finding Angle from Arc Length

Worked Example03

A sector has radius 10 cm and arc length 15 cm. Find the angle in:

(a) radians (b) degrees

Part (a) Rearrange

l = r θ

to get

θ = \frac{l}{r}

θ = \frac{15}{10} = 1.5 radians

Part (b) Convert to degrees

θ = 1.5 \times \frac{180}{π} = 85.9°

Sector Area Formula

Formula (in booklet)

A = \frac{1}{2} r^{2} θ

where

A

= sector area,

r

= radius,

θ

= angle in radians

Understanding the Formula

Sector area is a fraction of the full circle area:

A = \frac{θ}{2 π} \times π r^{2} = \frac{1}{2} r^{2} θ

Connection

Notice that

A = \frac{1}{2} r^{2} θ = \frac{1}{2} r \times r θ = \frac{1}{2} r l

So sector area = ½ × radius × arc length

Finding Sector Area

Worked Example04

A sector has radius 6 cm and angle

\frac{π}{3}

radians. Find the area of the sector, giving your answer in terms of π.

Step 1 Identify values:

r = 6

,

θ = \frac{π}{3}

Step 2 Apply formula:

A = \frac{1}{2} r^{2} θ

A = \frac{1}{2} \times 6^{2} \times \frac{π}{3} = \frac{1}{2} \times 36 \times \frac{π}{3} = 18 \times \frac{π}{3} = 6 π cm^{2}

Finding Radius from Sector Area

Worked Example05

A sector has angle 1.2 radians and area 37.5 cm². Find the radius.

Step 1 Write the formula:

A = \frac{1}{2} r^{2} θ

Step 2 Substitute:

37.5 = \frac{1}{2} \times r^{2} \times 1.2

Step 3 Solve for

r^{2}

:

37.5 = 0.6 r^{2}

r^{2} = \frac{37.5}{0.6} = 62.5

r = 62.5 = 7.91 cm

Segment of a Circle

Definition

A segment is the region between a chord and the arc it cuts off.

Segment Area Formula

Segment Area = Sector Area - Triangle Area

A = \frac{1}{2} r^{2} θ - \frac{1}{2} r^{2} sin θ = \frac{1}{2} r^{2} (θ - sin θ)

Note

The triangle is formed by the two radii and the chord. Its area is

\frac{1}{2} r^{2} sin θ

(using the area formula with included angle θ).

Finding Segment Area

Worked Example06

A chord subtends an angle of 1.2 radians at the centre of a circle of radius 10 cm. Find the area of the minor segment.

Step 1 Sector area:

A_{sector} = \frac{1}{2} \times 1 0^{2} \times 1.2 = 60 cm^{2}

Step 2 Triangle area:

A_{triangle} = \frac{1}{2} \times 1 0^{2} \times sin (1.2) = 50 \times 0.9320 = 46.60 cm^{2}

Step 3 Segment area:

A = 60 - 46.60 = 13.4 cm^{2}

(Or using combined formula:

A = \frac{1}{2} (10)^{2} (1.2 - sin 1.2) = 50 (1.2 - 0.932) = 13.4

)

Multi-step Problem

Worked Example07

A sector has perimeter 20 cm and area 21 cm². Find the radius and the angle (in radians).

Step 1 Set up equations

Perimeter:

r θ + 2 r = 20

→

r (θ + 2) = 20

... (1)

Area:

\frac{1}{2} r^{2} θ = 21

→

r^{2} θ = 42

... (2)

Step 2 From (1):

θ = \frac{20}{r} - 2

Step 3 Substitute into (2):

r^{2} (\frac{20}{r} - 2) = 42

20 r - 2 r^{2} = 42

2 r^{2} - 20 r + 42 = 0

→

r^{2} - 10 r + 21 = 0

(r - 3) (r - 7) = 0

→

r = 3

or

r = 7

Step 4 Find θ for each: If

r = 3

:

θ = \frac{20}{3} - 2 = \frac{14}{3}

rad; If

r = 7

:

θ = \frac{20}{7} - 2 = \frac{6}{7}

rad

Solutions:

r = 3

cm,

θ = \frac{14}{3}

rad OR

r = 7

cm,

θ = \frac{6}{7}

rad

Practice

1 A sector of a circle has radius 12 cm and angle

\frac{2 π}{3}

radians.

(a) Find the arc length, giving your answer in terms of π.

(b) Find the area of the sector, giving your answer in terms of π.

(c) Find the perimeter of the sector.

(a)

l = r θ = 12 \times \frac{2 π}{3} = 8 π

cm

(b)

A = \frac{1}{2} r^{2} θ = \frac{1}{2} \times 144 \times \frac{2 π}{3} = 72 \times \frac{2 π}{3} = 48 π

cm²

(c)

P = l + 2 r = 8 π + 24 = 8 π + 24

cm (or ≈ 49.1 cm)

Practice

2 The diagram shows a sector OAB of a circle with centre O and radius 8 cm. The angle AOB is 0.9 radians.

(a) Calculate the length of the arc AB.

(b) Calculate the area of the sector OAB.

(c) Calculate the area of the shaded segment.

(a)

l = r θ = 8 \times 0.9 = 7.2

cm

(b)

A_{sector} = \frac{1}{2} r^{2} θ = \frac{1}{2} \times 64 \times 0.9 = 28.8

cm²

(c) Triangle area:

A_{tri} = \frac{1}{2} \times 8^{2} \times sin (0.9) = 32 \times 0.7833 = 25.07

cm²

Segment area:

A = 28.8 - 25.07 = 3.73

cm² (or use

\frac{1}{2} r^{2} (θ - sin θ)

)