Terms	Radians	Degrees
Definition	Based on the radius of a circle; one radian is the angle subtended by an arc equal to the radius	Based on dividing a circle into 360 equal parts.
Range	A full circle is 2π radians	A full circle is 360°.
Mathematical Properties	Radians are often preferred in advanced mathematics because they simplify many formulas, especially in calculus and physics.	Degrees are more intuitive for everyday use and geometric visualizations
Occurrence	Radians arise naturally in many physical and mathematical contexts	Degrees are a human construct based on ancient Babylonian mathematics.
Periodicity of Trigonometric Functions	Trigonometric functions have a period of 2π.	The period is 360°.

Examples Related to Radian

Example 1: Convert 45° to radians.

Solution:

Formula: θ (in radians) = θ (in degrees) × (π / 180)

Calculation: 45 × (π / 180) = π/4 radians

45° = π/4 radians

Example 2: Convert 2 radians to degrees.

Solution:

Formula: θ (in degrees) = θ (in radians) × (180 / π)

Calculation: 2 × (180 / π) ≈ 114.59°

2 radians ≈ 114.59°

Example 3: The radius of a circle is 5 cm. What is the length of an arc that subtends an angle of π/3 radians at the center?

Solution:

Formula: s = r × θ, where s is arc length, r is radius, θ is angle in radians

Given: r = 5 cm, θ = π/3 radians

Calculation: s = 5 × (π/3) ≈ 5.24 cm

Arc length is approximately 5.24 cm

Example 4: How many radians are in a full circle?

Solution:

A full circle is 360^o

Using the conversion formula: 360 × (π / 180) = 2π radians

There are 2π radians in a full circle

Example 5: The minute hand of a clock moves through what angle in radians in 15 minutes?

Solution:

In 60 minutes, the minute hand moves through a full circle (2π radians)

In 15 minutes, it moves through 1/4 of this

Calculation: (1/4) × 2π = π/2 radians

Minute hand moves through π/2 radians in 15 minutes

Example 6: A wheel with a radius of 0.3 meters rotates through an angle of 4 radians. What distance does a point on the edge of the wheel travel?

Solution:

Formula: s = r × θ

Given: r = 0.3 m, θ = 4 radians

Calculation: s = 0.3 × 4 = 1.2 m

A point on the edge travels 1.2 meters

Example 7: Convert 5π/6 radians to degrees.

Solution:

Formula: θ (in degrees) = θ (in radians) × (180 / π)

Calculation: (5π/6) × (180 / π) = 150°

5π/6 radians = 150°

Example 8: What is the radian measure of a 30° angle?

Solution:

Formula: θ (in radians) = θ (in degrees) × (π / 180)

Calculation: 30 × (π / 180) = π/6 radians

30° = π/6 radians

Example 9: The arc length of a sector is 10 cm and the radius of the circle is 5 cm. What is the angle of the sector in radians?

Solution:

Formula: θ = s / r

Given: s = 10 cm, r = 5 cm

Calculation: θ = 10 / 5 = 2 radians

Angle of the sector is 2 radians

Example 10: If an angle of π/4 radians is subtended at the center of a circle of radius 8 cm, what is the area of the sector formed?

Solution:

Formula for sector area: A = (1/2) × r² × θ

Given: r = 8 cm, θ = π/4 radians

Calculation: A = (1/2) × 8² × (π/4) = 8π cm²

Area of sector is 8π cm² (approximately 25.13 cm²)

Practice Problem Related to Radian

Problem 1 :Convert 60° to radians.

Problem 2: Convert 5π/6 radians to degrees.

Problem 3: A wheel with a radius of 0.5 meters rotates through an angle of 3 radians. What distance does a point on the edge of the wheel travel?

Problem 4: The minute hand of a clock moves through what angle in radians in 20 minutes?

Problem 5: The arc length of a sector is 12 cm and the radius of the circle is 4 cm. What is the angle of the sector in radians?

Problem 6: If an angle of π/3 radians is subtended at the center of a circle of radius 6 cm, what is the area of the sector formed?

Problem 7: Convert 225° to radians, expressing the answer as a simplified fraction of π.

Problem 8: A central angle of 1.2 radians in a circle of radius 10 cm creates an arc. What is the length of this arc?

Problem 9: How many radians are there in 540°?

Problem 10: A sector of a circle has an area of 20 cm² and a radius of 5 cm. What is the central angle of this sector in radians?

Related Article:

• Radian Measure
• Differences between Radian and Degree
• Degree to Radians Calculator

FAQs on Radian

What is a radian?

A radian is a unit of angular measure defined as the angle subtended at the center of a circle by an arc equal in length to the radius.

How many radians are in a full circle?

There are 2π (approximately 6.28) radians in a full circle.

How do you convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180.

What is the radian measure of a right angle?

A right angle (90 degrees) is equal to π/2 radians.

Why are radians often preferred in mathematics?

Radians are preferred because they simplify many trigonometric formulas and are a more natural unit for calculus involving circular functions.