Radian Measure

• A radian is defined by the arc: 1 rad is the angle where arc length equals the radius. This means 2π rad = 360°.
• You can convert any angle using the anchor π = 180° — multiply or divide as needed.
• Arc length is given by s = rθ, where θ must be in radians.
• Area of a sector is A = ½r²θ — only the radius is squared, never θ.
• Area of a segment = Area of sector − Area of triangle = ½r²θ − ½r² sin θ.
• Certain angles have exact trig values — no decimals needed — derived from two special triangles.
• Exact values for 0°, 30°, 45°, 60°, 90°, 180° (and their radian equivalents) must be memorised.
• Values for angles like 135°, 240° etc. are found using graph symmetry plus the base exact values.

You already know degrees — a full rotation is 360°. But why 360? It's historical; the Babylonians liked that number. Radians are different: they're geometrically defined, which makes them the natural unit whenever circles, arcs, and calculus are involved.

Since the full circumference is 2πr, you can fit exactly 2π of those string-lengths around the circle. So a full rotation is 2π radians = 360°. Halve it: π radians = 180°. This one equation is your conversion key.

When you see an angle written with π and no degree symbol — like π/3 or 2π/5 — it is always in radians. No need to guess.

These are the angles that appear most often. Know them the same way you know multiplication tables — instant recall, no working needed.

For multiples — e.g. 3π/4 — use the base value: 3π/4 = 3 × (π/4) = 3 × 45° = 135°. For less common angles, just use π = 180° directly:

The arc length is the distance along the curved edge of a sector. Think of it as "unrolling" the curved part into a straight line.

This formula only works when θ is in radians. If given degrees, convert first. Answers should normally be left in terms of π unless told otherwise.

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