What is Radian?

Radian is defined as the link between a circle's radius and arc length. A basic unit of measurement for angles in mathematics. In mathematics, the radian is the unit of measurement for angles. The angle that forms at a circle's centre when the circumference's arc length equals the circle's radius is called a radian.

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

Definition of Radian

A radian is a unit of angular measurement used in mathematics and science. It is defined as the angle created when the length of the arc is equal to the radius of the circle.

There are 2π radians in a full circle, making it a natural way to measure angles. Radians are essential in trigonometry, calculus, and various fields of science, providing a direct link between linear and angular measurements.

Radian Formula

The formula relating radians to the radius and arc length of a circle is:

θ~=~\frac{s}{r}

where:

• θ (theta) is Angle in Radians
• s is Arc Length
• r is Radius of Circle

This formula essentially defines a radian: it's the angle formed when the arc length (s) is equal to the radius (r).

There are also formulas to convert between radians and degrees:

• To convert from degrees to radians: θ (in radians) = θ (in degrees) × (π / 180)
• To convert from radians to degrees: θ (in degrees) = θ (in radians) × (180 / π)

Remember that a full circle is:

• 360 degrees
• 2π radians (approximately 6.28 radians)

Conversion Between Degrees and Radians

Degrees to radians conversion is achieved using the formula:

Radians = Degrees × (π/180)

The reason for this is because 180° is equivalent to π radians. In essence, then, we are establishing a proportion.

Radians to degrees conversion is achieved using the formula:

Degrees = Radians × (180/π)

This operation is the opposite of the preceding one.

Examples of Degree to Radian Conversion

Some examples of degree to radian conversion:

• 45° to radians: 45 × (π/180) = π/4 ≈ 0.7854 radians
• 90° to radians: 90 × (π/180) = π/2 ≈ 1.5708 radians
• 180° to radians: 180 × (π/180) = π ≈ 3.1416 radians
• 270° to radians: 270 × (π/180) = 3π/2 ≈ 4.7124 radians
• 360° to radians: 360 × (π/180) = 2π ≈ 6.2832 radians

Examples of Radian to Degree Conversion

Some examples of radian to degree conversion:

• π/6 radians to degrees: (π/6) × (180/π) = 30°
• π/4 radians to degrees: (π/4) × (180/π) = 45°
• π/2 radians to degrees: (π/2) × (180/π) = 90°
• π radians to degrees: π × (180/π) = 180°
• 2π radians to degrees: 2π × (180/π) = 360°
• 1 radian to degrees: 1 × (180/π) ≈ 57.2958°

Radians and Degrees Table

Radian to Degree conversion table is added below:

Difference Between Radians and Degrees

Various differences between Radians and Degrees are added in the table below:

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