Converting Degrees to Radians- Formula and Example

What the Heck Is a Radian?

Most people learn degrees first. 90 degrees is a right angle. 180 is a straight line. Easy. Radians are different. They're based on the radius of a circle — the distance from center to edge. One radian is the angle you get when you take the radius and wrap it around the circumference of a circle. You can fit that radius about 6.283 times around the circle. That's 2π. Always. This is why math classes, physics, and engineering use radians instead of degrees. The math works out cleaner. Trigonometry functions expect radians. Calculus with angles? Radians only.

The Formula Nobody Remembers

Here's the conversion: Degrees to Radians: radians = degrees × (π / 180) That's it. Multiply by pi, divide by 180. Going the other way: Radians to Degrees: degrees = radians × (180 / π) Memorize the first one. The second one is just the reverse.

Quick Conversion Table

Degrees	Radians
0°	0
30°	π/6 ≈ 0.524
45°	π/4 ≈ 0.785
60°	π/3 ≈ 1.047
90°	π/2 ≈ 1.571
180°	π ≈ 3.142
270°	3π/2 ≈ 4.712
360°	2π ≈ 6.283

Learn these. They're used constantly.

How to Convert: Step by Step

Example 1: Convert 90° to radians

Take your degrees: 90
Multiply by π/180: 90 × (π/180)
Simplify: 90/180 = 1/2
Answer: π/2

That's 1.571 when you need the decimal.

Example 2: Convert 45° to radians

Start with 45
Multiply by π/180: 45 × (π/180)
Simplify: 45/180 = 1/4
Answer: π/4

Decimal form: 0.785

Example 3: Convert 3π/4 radians to degrees

Now reverse it:

Take your radians: 3π/4
Multiply by 180/π: (3π/4) × (180/π)
The πs cancel: 3 × 180 / 4
Answer: 540/4 = 135°

Why You Can't Just Use Degrees

Some reasons you need radians:

Calculus derivatives — d/dx(sin x) only works when x is in radians. The derivative of sin(θ) is cos(θ) only because θ is in radians. Degrees break this completely.
Physics formulas — Angular velocity, angular momentum, and most rotational mechanics use radians. ω = 2πf only works with radians.
Computer graphics — Game engines, animation libraries, and OpenGL expect radian inputs for rotation functions.
Engineering — Signal processing, control systems, and electrical engineering all use radians.

If you're writing code that rotates anything, you're using radians whether you know it or not.

Getting Started: Your First Conversion

Try these on paper:

Convert 120° to radians
Convert 5π/6 radians to degrees
Convert 1 radian to degrees (hint: about 57.3°)

Answers:

120 × π/180 = 2π/3
5π/6 × 180/π = 150°
1 × 180/π ≈ 57.3°

The third one is worth remembering: 1 radian ≈ 57.3 degrees. This comes up more than you'd expect.

Calculator Shortcuts

Most scientific calculators have a mode switch. Look for a button that says DRG or check your mode settings. Set it to RAD when working in radians, DEG when working in degrees. The calculator handles the conversion automatically. In Python:

import math
degrees = 90
radians = math.radians(degrees)  # Returns π/2

In JavaScript:

const degrees = 90;
const radians = degrees * (Math.PI / 180);

No built-in function? Use the formula. It never changes.

The Bottom Line

Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. That's the entire conversion. Everything else follows from those two formulas.