Convert angles between degrees and radians in both directions. See exact radian expressions as multiples of π for standard angles (30°, 45°, 60°, …), plus decimal approximations and step-by-step work.
Convert
Enter any angle in degrees (positive or negative).
Enter a decimal or use π notation: pi/3, 2pi/3, 3*pi/4
Quick Pick (Common Angles)
Results
Common Angles Reference
Click any row to load that angle.
| Degrees | Exact Radians | Decimal (rad) |
|---|
Why Radians?
1 radian = angle whose arc length = radius
A radian is defined by arc length: if you wrap a string of length r (the radius) along a circle's edge, the angle it sweeps out is exactly 1 radian.
Because the full circumference of a circle is 2πr, a complete revolution sweeps out exactly 2π radians. This gives us the fundamental conversion:
360° = 2π rad → 180° = π rad
From there: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
Radians simplify calculus (derivatives of sin/cos work cleanly) and are the standard in higher math and physics.
Unit Circle Connection
arc length = r · θ (θ in radians)
On the unit circle (radius = 1), the formula arc length = r·θ simplifies to arc length = θ. This means the radian measure of an angle directly equals the arc length along the unit circle.
That's why radians feel natural in trigonometry: when you say the angle is π/2, you're saying the arc from (1, 0) to (0, 1) is exactly π/2 units long — a quarter of the circumference (2π).
Key unit circle landmarks:
Work through the unit circle, radian measure, and all trig functions with a live tutor — step by step, at your pace.