Degrees to Radians Calculator

Convert between degrees and radians. Radians = degrees × π/180. Degrees = radians × 180/π. Includes common angle conversions and a visual reference for the unit circle.

📐 Trigonometry📐 rad = deg × π/180🔢 Math
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Formula & Reference

VariableSymbolFormulaUnits
Degrees to Radians Calculator—rad = deg × π/180rad or deg

Step-by-Step Examples

Example 1
90° to rad

90°×π/180=π/2≈1.5708 rad

  • Commonly needed: 90°=π/2, 180°=π, 270°=3π/2, 360°=2π
✓ π/2 radians
Example 2
π/3 to degrees

π/3 × 180/π = 60°

  • Exact value, no approximation needed
✓ 60°
Example 3
45° to rad

45°×π/180=π/4≈0.7854

  • 45°=π/4, sin(π/4)=cos(π/4)=√2/2
✓ π/4 radians

Real-World Applications

💻
Programming
All math libraries (Python, JS, C) use radians by default.
🧪
Physics
Angular velocity in rad/s, angular frequency in rad/s.
📐
Trigonometry
Unit circle and trig identities are cleaner in radians.
🏗️
Engineering
Structural analysis, rotational mechanics use radians.

Common Mistakes to Avoid

⚠️
Programming languages use radians

Math.sin(90) in JavaScript gives sin(90 radians), NOT sin(90°). Always convert first.

⚠️
π is irrational

Radian conversions of 'nice' degree values give irrational decimal approximations. Keep exact form (π/4) when possible.

⚠️
Gradians are a third system

Gradians: 400 grad = full circle. Rarely used outside surveying. Not the same as degrees or radians.

Frequently Asked Questions

Why are radians natural? ▾
Arc length = radius × angle in radians. This makes calculus of trig functions clean: d/dx[sin(x)] = cos(x) only in radians.
Common conversions to memorize ▾
0°=0, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 270°=3π/2, 360°=2π.
What is angular velocity? ▾
ω = Δθ/Δt in radians per second. v = ωr relates linear and angular speed. Converting to degrees would break this clean relationship.
Why 360 degrees in a circle? ▾
Historical: ancient Babylonians used base-60 arithmetic. 360 is highly composite (many factors), useful for dividing circles into equal parts.
What are gradians? ▾
Also called gon or grade. 100 gradians = right angle, 400 gradians = full circle. Used in French land surveying. Rarely seen today.

Related Math Calculators

📐 Trig Functions📐 Inverse Trig📐 Unit Circle📐 Arc Length📐 Angle

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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