What is Circular Measure (Radian)?

In mathematics, angles can be measured in different ways. The two most commonly used units for measuring angles are degrees and radians. The measurement of an angle in radians is called its circular measure.

Introduction

The concept of circular measure is especially important in trigonometry, geometry, calculus, and problems involving circles. In this article, we will learn what is circular measure, how radians are defined, the circular measure formula, the relationship between degrees and radians, and how to solve basic problems involving circular measure.

What is Circular Measure?

The circular measure of an angle is the measure of that angle expressed in radians.

A radian is defined using a circle. Consider a circle with centre O and radius r. If an angle at the centre intercepts an arc whose length is equal to the radius of the circle, then that angle is called one radian.

Therefore:

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

The symbol commonly used for radians is rad.

For example:

1 radian = 1 rad

Definition of Radian

A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of that circle.

If:

• Arc length = r
• Radius = r

then:

θ = s/r = r/r = 1 radian

where:

• θ = angle in radians
• s = length of the arc
• r = radius of the circle

Circular Measure Formula

The basic circular measure formula is:

θ = s/r

where:

θ = circular measure of the angle in radians
s = length of the corresponding arc
r = radius of the circle

This formula can also be rearranged to find arc length:

s = rθ

This is one of the most important formulas used in circular measure.

Why is Circular Measure Called Radian Measure?

Circular measure is called radian measure because it is based on the relationship between the arc length and the radius of a circle.

Unlike degree measurement, which divides a complete revolution into 360 equal parts, radian measurement is based directly on the geometry of a circle.

For this reason, radians are particularly useful in advanced mathematics.

Relation Between Degrees and Radians

We know that one complete revolution around a circle is:

360°

The circumference of a circle is:

2πr

Using the formula:

θ = s/r

for one complete revolution:

θ = 2πr/r

Therefore:

θ = 2π radians

Hence:

360° = 2π radians

Dividing both sides by 2:

180° = π radians

This is the most important relationship between degrees and radians.

Therefore:

180° = π rad

Degree to Radian Conversion Formula

To convert an angle from degrees into radians, use:

Angle in radians = Angle in degrees × π/180

Example 1: Convert 60° into radians

Using:

θ = 60 × π/180

θ = π/3

Therefore:

60° = π/3 radians

Example 2: Convert 90° into radians

θ = 90 × π/180

θ = π/2

Therefore:

90° = π/2 radians

Example 3: Convert 45° into radians

θ = 45 × π/180

θ = π/4

Therefore:

45° = π/4 radians

Radian to Degree Conversion Formula

To convert radians into degrees, use:

Angle in degrees = Angle in radians × 180/π

Example 4: Convert π/6 radians into degrees

θ = π/6 × 180/π

Cancel π:

θ = 180/6

θ = 30°

Therefore:

π/6 radians = 30°

Example 5: Convert 3π/4 radians into degrees

θ = 3π/4 × 180/π

Cancel π:

θ = 3 × 180/4

θ = 135°

Therefore:

3π/4 radians = 135°

Common Angles in Degrees and Radians

Students should remember the following important conversions:

Angle in Degrees	Angle in 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

These values are frequently used in trigonometry.

Circular Measure Using Arc Length

The circular measure of an angle can be calculated when the radius of the circle and the length of the corresponding arc are known.

The formula is:

θ = s/r

Example 6

A circle has a radius of 7 cm and an arc has a length of 14 cm. Find the circular measure of the angle.

Given:

r = 7 cm

s = 14 cm

Using:

θ = s/r

θ = 14/7

θ = 2 radians

Therefore, the circular measure of the angle is:

2 radians

Finding Arc Length from Circular Measure

If the angle is given in radians, the length of the arc can be calculated using:

s = rθ

Example 7

Find the length of an arc when the radius is 10 cm and the angle is π/3 radians.

Given:

r = 10 cm

θ = π/3

Using:

s = rθ

s = 10 × π/3

Therefore:

s = 10π/3 cm

If π is taken as 3.142:

s ≈ 10.47 cm

Important Facts About Circular Measure

1. Circular measure is the measure of an angle in radians.
2. One complete revolution is 2π radians.
3. A straight angle is π radians.
4. A right angle is π/2 radians.
5. One radian is approximately 57.3°.
6. The formula for circular measure is θ = s/r.
7. The formula for arc length is s = rθ.
8. The radius and arc length must be expressed in the same unit when using θ = s/r.
9. Radian measure is widely used in trigonometry and calculus.
10. Radians are dimensionless because both arc length and radius have the same unit.

Difference Between Degree Measure and Circular Measure

Degree Measure	Circular Measure
Angle is measured in degrees	Angle is measured in radians
Complete revolution = 360°	Complete revolution = 2π radians
Straight angle = 180°	Straight angle = π radians
Right angle = 90°	Right angle = π/2 radians
Based on division of a circle into 360 parts	Based on arc length and radius

What is Circular Measure Used For?

Circular measure is useful in many areas of mathematics.

It is used in:

• Trigonometry
• Geometry
• Arc-length calculations
• Sector-area calculations
• Calculus
• Circular motion
• Physics and engineering

Radians are particularly important because many mathematical formulas become simpler when angles are expressed in radians.

Circular Measure and Sector Area

Circular measure is also used to calculate the area of a sector.

When the angle θ is measured in radians, the area of a sector is:

A = 1/2 r²θ

where:

A = area of the sector
r = radius
θ = angle in radians

Example 8

Find the area of a sector with radius 6 cm and angle π/3 radians.

Using:

A = 1/2 r²θ

A = 1/2 × 6² × π/3

A = 1/2 × 36 × π/3

A = 6π cm²

Therefore, the area of the sector is:

6π cm²

Is a Circular Measuring Tool the Same as Circular Measure?

The terms circular measure and circular measuring tool should not be confused.

Circular measure is a mathematical method of measuring an angle in radians.

A measuring tool, such as a protractor, is a physical instrument used to measure angles. A normal protractor generally measures angles in degrees, although some specialized instruments can provide other angular measurements.

Therefore, circular measure is a mathematical concept, not the name of a physical measuring instrument.

Circular Measure as a Level of Angle Measurement

The phrase circular measure as level may sometimes appear in searches or educational material. In mathematics, circular measure refers specifically to expressing an angle in radians. It is not a separate “level” of an angle; rather, it is another method of representing the magnitude of an angle.

For example:

180° = π radians

Both expressions represent exactly the same angle, but one uses degrees while the other uses circular/radian measure.

Quick Method for Degree-Radian Conversion

Students can remember these two formulas:

Degrees to Radians

Multiply by π/180

For example:

120° × π/180 = 2π/3

Radians to Degrees

Multiply by 180/π

For example:

2π/3 × 180/π = 120°

This simple rule makes conversion between degrees and radians easy.

Solved Questions on Circular Measure

Question 1

Convert 150° into radians.

Solution:

150 × π/180

= 5π/6

Answer: 5π/6 radians

Question 2

Convert 2π/3 radians into degrees.

Solution:

2π/3 × 180/π

= 120°

Answer: 120°

Question 3

Find the circular measure of an angle that subtends an arc of 15 cm in a circle of radius 5 cm.

Solution:

θ = s/r

θ = 15/5

θ = 3 radians

Answer: 3 radians

Question 4

Find the length of an arc of a circle with radius 8 cm if the angle at the centre is π/4 radians.

Solution:

s = rθ

s = 8 × π/4

s = 2π cm

Answer: 2π cm

Summary

The circular measure of an angle is its measurement in radians. A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of that circle.

The most important formulas are:

θ = s/r

s = rθ

The fundamental conversion relationship is:

180° = π radians

For degree-to-radian conversion:

Degrees × π/180

For radian-to-degree conversion:

Radians × 180/π

Understanding circular measure is essential for solving problems involving arcs, sectors, trigonometry, and many advanced mathematical concepts.

For more mathematics notes, recorded lectures, assessments, lesson plans, and Class 9 Maths learning resources, students can explore Nisar Math Academy.

Short Notes on Circular Measure

Definition

Circular measure is the measurement of an angle in radians.

Radian

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

Main Formula

θ = s/r

Arc Length Formula

s = rθ

Important Relation

180° = π radians

Complete Revolution

360° = 2π radians

Right Angle

90° = π/2 radians

Straight Angle

180° = π radians

Degree to Radian

θ(rad) = θ(degree) × π/180

Radian to Degree

θ(degree) = θ(rad) × 180/π

Sector Area

A = 1/2 r²θ

where θ must be in radians.

MCQs on Circular Measure

1. What is the circular measure of an angle?

A. Its measurement in metres
B. Its measurement in radians
C. Its measurement in centimetres
D. Its measurement in square units

Answer: B. Its measurement in radians

2. One complete revolution is equal to:

A. π radians
B. π/2 radians
C. 2π radians
D. 4π radians

Answer: C. 2π radians

3. 180° is equal to:

A. π/2 radians
B. π radians
C. 2π radians
D. 3π radians

Answer: B. π radians

4. The formula for circular measure is:

A. θ = r/s
B. θ = sr
C. θ = s/r
D. θ = s + r

Answer: C. θ = s/r

5. If the arc length is equal to the radius, the angle is:

A. 0 radians
B. 1 radian
C. 2 radians
D. π radians

Answer: B. 1 radian

6. 90° is equal to:

A. π/4 radians
B. π/2 radians
C. π radians
D. 2π radians

Answer: B. π/2 radians

7. 60° in radians is:

A. π/2
B. π/3
C. π/4
D. π/6

Answer: B. π/3

8. π/4 radians are equal to:

A. 30°
B. 45°
C. 60°
D. 90°

Answer: B. 45°

9. The formula for the length of an arc is:

A. s = rθ
B. s = r/θ
C. s = θ/r
D. s = r + θ

Answer: A. s = rθ

10. A circle has radius 5 cm and an arc length of 10 cm. Its circular measure is:

A. 1 radian
B. 2 radians
C. 5 radians
D. 10 radians

Answer: B. 2 radians

11. 360° is equal to:

A. π radians
B. 2π radians
C. π/2 radians
D. 3π/2 radians

Answer: B. 2π radians

12. Which unit is normally used for circular measure?

A. Degree
B. Radian
C. Metre
D. Centimetre

Answer: B. Radian

13. 3π/4 radians are equal to:

A. 90°
B. 120°
C. 135°
D. 180°

Answer: C. 135°

14. If r = 4 cm and θ = π/2 radians, the arc length is:

A. 2π cm
B. 4π cm
C. 8π cm
D. π/2 cm

Answer: A. 2π cm

15. Which formula is used for the area of a sector when θ is in radians?

A. A = rθ
B. A = 1/2 r²θ
C. A = 2rθ
D. A = r²/θ

Answer: B. A = 1/2 r²θ

Worksheet / Assignment: Circular Measure

Part A: Fill in the Blanks

1. Circular measure is expressed in ________.
2. One complete revolution is equal to ________ radians.
3. 180° is equal to ________ radians.
4. The formula for circular measure is θ = ________.
5. The formula for arc length is s = ________.
6. 90° is equal to ________ radians.
7. 45° is equal to ________ radians.
8. π radians are equal to ________ degrees.

Part B: Convert the Following Degrees into Radians

1. 30°
2. 45°
3. 60°
4. 90°
5. 120°
6. 150°
7. 180°
8. 270°

Part C: Convert the Following Radians into Degrees

1. π/6
2. π/4
3. π/3
4. π/2
5. 2π/3
6. 3π/4
7. 5π/6
8. 2π

Part D: Solve the Following

1. Find the circular measure of an angle that subtends an arc of 12 cm in a circle with radius 4 cm.
2. Find the length of an arc when the radius is 7 cm and the angle is π/2 radians.
3. Find the length of an arc when the radius is 10 cm and the angle is π/5 radians.
4. A circle has radius 8 cm and an arc length of 16 cm. Find the circular measure of the angle.
5. Find the area of a sector having radius 6 cm and central angle π/3 radians.

Part E: Conceptual Questions

1. Define circular measure.
2. What is one radian?
3. Why is circular measure also called radian measure?
4. State the relationship between degrees and radians.
5. Write the formula used to convert degrees into radians.
6. Write the formula used to convert radians into degrees.
7. What is the difference between degree measure and circular measure?
8. Why must the angle be in radians when using the formula s = rθ?

Answer Key

Part A

1. radians
2. 2π
3. π
4. s/r
5. rθ
6. π/2
7. π/4
8. 180°

Part B

1. π/6
2. π/4
3. π/3
4. π/2
5. 2π/3
6. 5π/6
7. π
8. 3π/2

Part C

1. 30°
2. 45°
3. 60°
4. 90°
5. 120°
6. 135°
7. 150°
8. 360°

Part D

1. θ = 12/4 = 3 radians
2. s = 7 × π/2 = 7π/2 cm
3. s = 10 × π/5 = 2π cm
4. θ = 16/8 = 2 radians
5. A = 1/2 × 6² × π/3 = 6π cm²

Key Takeaways

• Circular measure = angle measured in radians.
• 1 radian is subtended by an arc equal to the radius.
• 180° = π radians
• 360° = 2π radians
• θ = s/r
• s = rθ
• A = 1/2 r²θ
• To convert degrees to radians, multiply by π/180.
• To convert radians to degrees, multiply by 180/π.
• Radian measure is especially important in trigonometry, geometry, and calculus.

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