3.2 Radians and Degrees

What is a radian?

A radian measures angles based on arc length: it's the angle subtended when the arc length equals the radius of the circle. One full revolution equals 2π radians = 360°. Since radians relate directly to arc length, they're the natural unit for calculus and periodic functions.

Converting between degrees and radians

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Key benchmarks: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. Memorizing these speeds up work with the unit circle.

Why radians matter

Radians connect angle measure directly to arc length (s = rθ) and area of a sector (A = ½r²θ), with θ in radians. This makes formulas cleaner than with degrees, which is why radians are the standard unit in higher math, physics, and calculus-based reasoning about periodic functions.

Key terms

Radian (definition)

The angle subtended by an arc whose length equals the circle's radius; 2π radians = one full revolution.

Degrees → Radians conversion

Multiply degrees by π/180.

Radians → Degrees conversion

Multiply radians by 180/π.

Arc length formula

s = rθ, where θ is in radians and r is the radius.

Key benchmark angles

30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

Try an AP-style question

Convert 150° to radians.

1. 2π/3
2. 5π/6
3. 3π/4
4. 5π/3