Degree to Radian Conversion- Formula and Examples

What Is Radian Measure and Why It Exists

Degrees are arbitrary. Seriously—they come from the ancient Babylonians who thought 360 was a nice round number (divisible by everything). Radians are mathematically natural. One radian is the angle where the arc length equals the radius of the circle.

Most math textbooks, calculus classes, and engineering work uses radians. If you're writing code, building anything with physics, or taking higher-level math, you'll need to convert between the two. Here's how.

The Conversion Formula

The relationship between degrees and radians:

π radians = 180 degrees

From this, you get your conversion formulas:

Degrees to Radians: multiply by (π/180)

Radians to Degrees: multiply by (180/π)

That's it. That's the whole formula.

How to Convert Degrees to Radians

Take your degree value. Multiply by π. Divide by 180. Done.

Or in calculator terms: degrees × 0.0174533 = radians

The number 0.0174533 is just π/180 pre-computed. Use whichever feels faster.

Step-by-Step Example

Convert 45° to radians:

45 × (π/180) = π/4

In decimal form: 45 × 0.0174533 = 0.7854 radians

Check: π/4 ≈ 3.1416/4 = 0.7854 ✓

More Examples

90 degrees to radians:

90 × (π/180) = π/2 ≈ 1.5708 radians

30 degrees to radians:

30 × (π/180) = π/6 ≈ 0.5236 radians

120 degrees to radians:

120 × (π/180) = 2π/3 ≈ 2.0944 radians

360 degrees to radians:

360 × (π/180) = 2π ≈ 6.2832 radians

See the pattern? Simple multiplication every time.

Quick Conversion Table

Save this. You'll use it constantly.

Converting Radians to Degrees

Going the other way? Just reverse it.

radians × (180/π) = degrees

Or: radians × 57.2958 = degrees

Example: Convert 2.5 radians to degrees

2.5 × 57.2958 = 143.24°

How to Do This in Code

Most programming languages have built-in conversion functions. Here's the deal:

• Python: Use math.radians(degrees) or math.degrees(radians)
• JavaScript: Multiply by (Math.PI / 180) or (180 / Math.PI)
• Excel/Sheets: Use RADIANS(degrees) or DEGREES(radians)

No need to memorize formulas when coding—just call the function.

Why This Matters

Calculus derivatives and integrals with trig functions only work in radians. Put in 90 degrees where the formula expects radians and you'll get wrong answers.

Physics, engineering, computer graphics, signal processing—all use radians because the math is cleaner. The formulas are shorter. The derivatives are simpler.

Know both. Convert between them without thinking. That's the goal.

The Simple Way to Remember

Forget memorizing formulas. Remember this:

Divide degrees by 180, multiply by π.

That's the whole conversion in one sentence.

Now you can stop reading. Convert anything you need.