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Radians and degrees
Beginner
2π radians is one full turn.
180° = π radA radian is the angle for which arc length equals the radius; 2π rad = 360°. Formulas such as v = ωr and s = rθ need radians.
90° = π/2 rad ≈ 1.57.
Calculators must be in the right mode.
In more detail
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. There are 2π radians in a full turn, and radians make rotational formulas simple.
Real-world uses
- Engineers use radians in formulas for gears and rotating machinery.
- Computer graphics libraries use radians for rotation angles.
- A protractor reads degrees, but calculus and physics formulas need radians, so conversion is needed before using them.
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Frequently asked questions
How do I convert degrees to radians?
Multiply by π divided by 180.
Why not use degrees?
Formulas such as v = rω need radians to hold without extra constants.
Why does 2π radians mean one full turn?
The circumference of a circle is 2πr, so the angle for a full arc of length 2πr on a radius r is 2π radians.
Related topics in Rotational Motion
Angular displacement · Angular velocity · Angular acceleration · Relation between linear and angular quantities · Rotational kinematics · Torque