Radians or Degrees Calculator

Radians and degrees are two common units for measuring angles. While degrees are more familiar in everyday contexts, radians are often used in advanced mathematics and physics. This calculator helps you convert between these units quickly and accurately.

What Are Radians and Degrees?

Angles can be measured in two primary units: degrees and radians. Each has its own advantages depending on the context.

Degrees

Degrees are the most commonly used unit for measuring angles. A full circle is divided into 360 degrees. This system is intuitive for everyday use, such as in navigation or construction.

Radians

Radians are a unit of angular measurement based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. A full circle is 2π radians (approximately 6.283 radians).

Key Relationship

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

1 degree = (π/180) radians ≈ 0.01745 radians

How to Convert Between Radians and Degrees

Converting between radians and degrees is straightforward once you understand the relationship between the two units.

Degrees to Radians

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

Formula

radians = degrees × (π/180)

Radians to Degrees

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

Formula

degrees = radians × (180/π)

Example Conversion

Let's convert 90 degrees to radians:

90° × (π/180) ≈ 1.5708 radians

This is equivalent to π/2 radians.

When to Use Radians vs. Degrees

The choice between radians and degrees depends on the context and the mathematical operations being performed.

Using Degrees

Everyday applications like navigation, construction, and basic geometry
When working with angles that are easily expressed in fractions of a circle (e.g., 90°, 180°)
In fields where degrees are the standard (e.g., surveying, architecture)

Using Radians

Advanced mathematics, physics, and engineering
Calculus, where derivatives and integrals of trigonometric functions are simpler with radians
When working with circular motion or wave functions

Note

Most scientific calculators have a mode switch between degrees and radians. Ensure your calculator is set to the correct mode before performing calculations.

Common Angles in Both Units

Here's a table showing common angles in both degrees and radians:

Degrees	Radians	Description
0°	0	No rotation
30°	π/6	One sixth of a circle
45°	π/4	Diagonal of a square
60°	π/3	Equilateral triangle angle
90°	π/2	Right angle
180°	π	Straight angle
270°	3π/2	Three-quarters of a circle
360°	2π	Full circle

FAQ

Why do we need both radians and degrees?

Degrees are more intuitive for everyday use, while radians simplify complex mathematical operations, especially in calculus and physics. The choice depends on the context and the specific problem being solved.

How do I know if my calculator is set to degrees or radians?

Most scientific calculators have a mode display that shows "DEG" for degrees and "RAD" for radians. If you're unsure, check the manual or consult the calculator's documentation.

Can I convert radians to degrees without using π?

No, the conversion between radians and degrees inherently involves π because the two units are defined based on the circle's properties. You cannot convert between them without using π or its approximate value.

Are radians always smaller than degrees?

Yes, because a full circle is 360 degrees but only approximately 6.283 radians. Therefore, any angle measured in radians will be smaller than its degree counterpart for the same rotation.