Radians

What is a radian?

Imagine wrapping the radius of a circle around its circumference. The angle at the centre that cuts off an arc equal in length to the radius is 1 radian.

On a full circle: Circumference = 2πr, so there are 2π radians in a full turn.

360° = 2π radians ⇒ 180° = π radians

This gives the conversion formulas: θ (radians) = θ (degrees) × π / 180 and θ (degrees) = θ (radians) × 180 / π.

Interactive Radian Wheel

Drag the red point around the circle, or use the slider. The angle is shown in both degrees and radians. When you get close to special angles, it gently snaps and shows the exact value.

Hybrid drag: free movement with gentle snapping to π/6, π/4, π/3, π/2, … (toggle below).

Angle θ (degrees)

Snap to special angles (π/6, π/4, π/3, π/2, …)

Angle: °

Angle in radians:

Conversions:

θ (radians) = θ° × π / 180

θ (degrees) = θ (radians) × 180 / π

Teacher mode (show exact forms where possible)

Example

Convert between degrees and radians

1. Degrees → radians

Convert 135° to radians.

Use θ (radians) = θ° × π / 180.
θ = 135 × π / 180 = (3π/4).

135° = 3π/4 radians

2. Radians → degrees

Convert 5π/6 to degrees.

Use θ (degrees) = θ (radians) × 180 / π.
θ = (5π/6) × 180 / π = 5 × 30 = 150°.

5π/6 radians = 150°

Try it yourself

Click “New angle” to generate a random angle. Decide whether to convert to degrees or radians, then reveal the answer.

Given:

Convert to: