Angle Degrees Radians Calculator

Convert between degrees and radians with this precise angle calculator. Learn the conversion formulas, see practical examples, and understand when to use each unit in calculations.

What is Angle Conversion?

Angle conversion is the process of changing an angle measurement from degrees to radians or vice versa. Degrees and radians are both units of angular measurement, but they are used in different contexts and have different mathematical properties.

Degrees are commonly used in everyday contexts like navigation, construction, and geometry. Radians are more commonly used in advanced mathematics, physics, and engineering because they provide a more natural representation of angles in circular functions.

Key difference: A full circle is 360 degrees or 2π radians. This means the conversion factor between degrees and radians is π/180.

Degrees to Radians Formula

To convert degrees to radians, use the following formula:

Radians = Degrees × (π / 180)

This formula works because a full circle (360 degrees) is equivalent to 2π radians. Therefore, to convert degrees to radians, you multiply the degree measurement by π/180.

Example: Convert 90 degrees to radians

Using the formula:

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

So, 90 degrees is equal to π/2 radians or approximately 1.5708 radians.

Radians to Degrees Formula

To convert radians to degrees, use the following formula:

Degrees = Radians × (180 / π)

This formula works because a full circle (2π radians) is equivalent to 360 degrees. Therefore, to convert radians to degrees, you multiply the radian measurement by 180/π.

Example: Convert π/4 radians to degrees

Using the formula:

Degrees = (π/4) × (180/π) = 45 degrees

So, π/4 radians is equal to 45 degrees.

Common Angle Conversions

Here are some common angle conversions between degrees and radians:

Degrees	Radians	Description
0°	0	No rotation
30°	π/6	Common angle in right triangles
45°	π/4	Diagonal of a square
60°	π/3	Common angle in equilateral triangles
90°	π/2	Right angle
180°	π	Straight angle
270°	3π/2	Three-quarters of a circle
360°	2π	Full circle

When to Use Degrees vs Radians

Choosing between degrees and radians depends on the context of your calculations:

Use Degrees When:

Working with everyday measurements (e.g., navigation, construction)
Using trigonometric functions in basic geometry
Reading angles from protractors or compasses

Use Radians When:

Working with calculus and advanced mathematics
Using trigonometric functions in physics and engineering
Working with circular functions and periodic phenomena

Most scientific calculators have a "degree" and "radian" mode to switch between units as needed.

FAQ

Why do we need both degrees and radians?

Degrees and radians serve different purposes. Degrees are more intuitive for everyday measurements, while radians provide a more natural representation of angles in circular functions and calculus. Most scientific calculators support both units.

How do I convert degrees to radians?

Multiply the degree measurement by π/180. For example, 90 degrees × (π/180) = π/2 radians.

How do I convert radians to degrees?

Multiply the radian measurement by 180/π. For example, π/4 radians × (180/π) = 45 degrees.

What is the relationship between degrees and radians?

A full circle is 360 degrees or 2π radians. This means the conversion factor between degrees and radians is π/180.