Degrees vs Radians Calculator

Degrees and radians are two common units for measuring angles. While degrees are more familiar in everyday life, radians are widely used in advanced mathematics and physics. This guide explains the difference between degrees and radians, how to convert between them, and when to use each unit.

What are Degrees and Radians?

An angle is a measure of rotation between two rays with a common endpoint. The most common units for measuring angles are degrees and radians.

Degrees

Degrees are a unit of measurement for angles. A full circle is divided into 360 degrees. This system is based on the ancient Babylonian division of the circle into 360 parts.

Degrees are commonly used in everyday contexts, such as navigation, construction, and geography. For example, a right angle is 90 degrees, and a straight angle is 180 degrees.

Radians

Radians are another unit of measurement for angles. A radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

1 radian = (180/π) degrees ≈ 57.2958 degrees

Radians are used extensively in advanced mathematics, physics, and engineering because they simplify many calculations involving circular motion and periodic functions.

How to Convert Degrees to Radians

To convert degrees to radians, use the following formula:

Radians = Degrees × (π/180)

For example, to convert 90 degrees to radians:

Radians = 90 × (π/180) = π/2 ≈ 1.5708 radians

This conversion is useful in trigonometric functions, where radians are often the preferred unit.

How to Convert Radians to Degrees

To convert radians to degrees, use the following formula:

Degrees = Radians × (180/π)

For example, to convert π/2 radians to degrees:

Degrees = (π/2) × (180/π) = 90 degrees

This conversion is useful when working with angles in a context where degrees are more familiar.

Common Angle Conversions

Here are some common angle conversions between degrees and radians:

Degrees	Radians
0°	0
30°	π/6 ≈ 0.5236
45°	π/4 ≈ 0.7854
60°	π/3 ≈ 1.0472
90°	π/2 ≈ 1.5708
180°	π ≈ 3.1416
270°	3π/2 ≈ 4.7124
360°	2π ≈ 6.2832

This table provides quick reference points for common angle conversions.

When to Use Degrees vs Radians

Choosing between degrees and radians depends on the context and the specific application.

Use Degrees When:

Working with everyday measurements, such as navigation or construction.
Using trigonometric functions in a context where degrees are more familiar.
Dealing with angles that are easily expressed in multiples of 30° or 45°.

Use Radians When:

Working with advanced mathematics, physics, or engineering.
Using trigonometric functions in calculus or differential equations.
Dealing with circular motion or periodic functions.

In many scientific and engineering contexts, radians are the preferred unit because they simplify calculations involving circular motion and periodic functions.

FAQ

What is the difference between degrees and radians?

Degrees and radians are both units for measuring angles. A full circle is 360 degrees or 2π radians. Degrees are more familiar in everyday contexts, while radians are used in advanced mathematics and physics.

How do I convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180. For example, 90 degrees is π/2 radians.

How do I convert radians to degrees?

To convert radians to degrees, multiply the radian value by 180/π. For example, π/2 radians is 90 degrees.

When should I use degrees instead of radians?

Use degrees when working with everyday measurements or in contexts where degrees are more familiar. Use radians in advanced mathematics, physics, and engineering.

What is the relationship between degrees and radians?

Degrees and radians are related by the formula: 1 radian = 180/π degrees ≈ 57.2958 degrees. This relationship allows for easy conversion between the two units.