Convert Angle From Degrees to Radians Calculator

Converting angles between degrees and radians is a fundamental skill in mathematics, physics, and engineering. This calculator provides an easy way to perform this conversion, along with explanations of when and why you might need to use each unit.

Conversion Formula

The relationship between degrees and radians is defined by the formula:

Radians = Degrees × (π / 180)

Conversely, to convert radians to degrees:

Degrees = Radians × (180 / π)

This conversion is based on the fact that a full circle is 360 degrees or 2π radians. Therefore, π radians equals 180 degrees.

π (pi) is approximately 3.141592653589793, but most calculations use the precise mathematical constant rather than this approximation.

When to Use Degrees vs Radians

Degrees and radians are used in different contexts based on the application:

Degrees: Commonly used in everyday applications, geography, and navigation. Most people are more familiar with degrees, especially for angles less than 360°.
Radians: Preferred in advanced mathematics, physics, and engineering, particularly when working with calculus, trigonometric functions, and circular motion.

For example, when measuring angles in a circle, radians provide a more natural unit because the circumference of a unit circle is 2π radians, which corresponds to 360 degrees.

Common Angle Conversions

Here are some common angle conversions between degrees and radians:

Degrees	Radians	Description
0°	0	No rotation
30°	π/6 ≈ 0.5236	Common in trigonometric calculations
45°	π/4 ≈ 0.7854	Diagonal of a square
60°	π/3 ≈ 1.0472	Equilateral triangle angle
90°	π/2 ≈ 1.5708	Right angle
180°	π ≈ 3.1416	Straight angle
270°	3π/2 ≈ 4.7124	Three-quarters of a circle
360°	2π ≈ 6.2832	Full circle

Practical Examples

Let's look at some practical examples of converting between degrees and radians:

Example 1: Converting 45° to Radians

Using the formula:

Radians = 45° × (π / 180) ≈ 45 × 0.01745 ≈ 0.7854 radians

This is useful when calculating the arc length or area of a sector in a circle.

Example 2: Converting 2 Radians to Degrees

Using the formula:

Degrees = 2 × (180 / π) ≈ 2 × 57.2958 ≈ 114.5916°

This might be used when interpreting angular velocity measurements in physics.

Example 3: Full Circle Conversion

A full circle is 360° or 2π radians. This equivalence is fundamental in trigonometry and calculus.

Frequently Asked Questions

Why do we need to convert between degrees and radians?

Different fields use different units for angles. Degrees are more intuitive for everyday measurements, while radians are more mathematically convenient for advanced calculations involving calculus and circular functions.

Is π really 3.141592653589793?

Yes, π is an irrational number that continues infinitely without repeating. For most practical purposes, 3.141592653589793 is precise enough, but mathematical calculations typically use the exact value of π.

Can I use this calculator for angles larger than 360°?

Yes, the calculator will work for any angle value. However, angles larger than 360° or 2π radians represent multiple full rotations around a circle.

What's the difference between degrees and radians in terms of trigonometric functions?

Trigonometric functions (sine, cosine, tangent) have different periods when using degrees versus radians. For degrees, the period is 360°, while for radians, the period is 2π. This affects how the functions repeat and their values at specific points.