What is a radian?
A radian is the angle you get when the arc along a circle equals the circle's radius. A full turn is 2π radians, which is why 360° = 2π and 180° = π radians.
// maths › Trigonometric Functions & Graphs
Degrees & radians
Convert angles between degrees and radians in either direction, with the radian answer shown as a multiple of pi where it is exact, illustrated by a dual arc where both measures update together around the circle.
radians = degrees × π/180 · degrees = radians × 180/π
Frequently asked questions
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example, 90° × π/180 = π/2 radians.
Can you give a worked example?
To convert 180° to radians: 180 × π/180 = π ≈ 3.1416 radians. Going the other way, π/3 radians × 180/π = 60°.
Why use radians instead of degrees?
Radians tie an angle directly to arc length and make the formulas of calculus and wave physics much simpler. Degrees are convenient for everyday angles, but advanced maths and science almost always use radians.
Where is this used in real life?
Engineers and physicists use radians for rotational motion, angular velocity, and oscillations; computer graphics and robotics use them for rotation; and they are standard in signal processing and astronomy.
How do I find arc length from an angle?
Use s = rθ, where r is the radius and θ is the angle in radians. For example, a radius of 5 with θ = 2 radians gives s = 5 × 2 = 10. If the angle is in degrees, convert it with ×π/180 first — switch the toggle and the calculator does this for you.
How do I find the area of a sector?
Use A = ½r²θ with θ in radians. A radius of 4 and θ = π/2 (i.e. 90°) gives A = ½ × 16 × π/2 ≈ 12.57 square units. The sector is the fraction θ/2π of the whole circle, which is where the formula comes from.