Circular Measure

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Cambridge (CIE) IGCSE — Additional Maths

Circular Measure

Big idea: Radians are just a different — and honestly smarter — way to measure angles, one that makes arc lengths and sector areas fall out of dead-simple formulas instead of messy fraction-of-360° calculations.

Summary

Radians measure angles using arc length instead of "360 slices" — 1 radian is the angle made when the arc equals the radius.
The golden conversion fact: π radians = 180°. Everything else builds from this.
An arc is a curved piece of the circumference; a sector is the pizza-slice shape (two radii + an arc).
In degrees: arc length l = (θ/360) × 2πr, sector area A = (θ/360) × πr².
In radians, these collapse beautifully to: l = rθ and A = ½r²θ.
Minor arc/sector = angle < 180°. Major arc/sector = angle > 180° (the "rest of the pizza").
Perimeters of sectors/compound shapes = arc length + straight radii (don't forget the straight bits!).
Always check your calculator is in the correct mode (degrees vs radians) before computing.

Topic 1 — Radian Measure

1What actually IS a radian?

Degrees are a completely arbitrary unit — someone decided a full turn is 360° a very long time ago (probably because 360 has loads of nice divisors). Radians, on the other hand, are built directly out of the geometry of the circle itself, which is why they make so many formulas simpler later on.

Analogy: Imagine you take a piece of string exactly as long as the radius of a circle, and you lay it along the circumference, starting from one point. Walk your finger along that string-length of arc. The angle you've swept out at the centre, from the start to where your string ends — that angle is defined as 1 radian. You're literally measuring the angle in "radius-lengths of arc."

Formally: 1 radian is the angle in a sector of radius 1 and arc length 1. A circle with radius 1 is called a unit circle, and it's the cleanest way to picture this — because in a unit circle, the arc length and the angle (in radians) are literally the same number.

In the unit circle, an arc of length 1 sweeps out an angle of exactly 1 radian (≈ 57.3°).

Notation note The symbol for radians is technically ᶜ (a small c), but in practice you'll almost always see it written as rad, or with no symbol at all when π is involved (since it's "obviously" radians). You must never drop the ° symbol for degrees though — that one's non-negotiable, since 3 and 3° mean very different things.

2Converting between radians and degrees

This is the single most important fact in the whole chapter — memorise it so hard you could recite it half-asleep:

The golden equation

π^c = 180°

π radians is exactly the same angle as 180° (a straight line / half a turn)

From this one fact, you can derive absolutely everything else. Want to convert radians → degrees? Multiply by 180/π. Want degrees → radians? Multiply by π/180.

Conversion rules

radians → degrees: × 180/π | degrees → radians: × π/180

It's genuinely worth memorising these common conversions outright, because they show up constantly and let you sanity-check your work instantly:

Radians	Degrees
2π	360°
π	180°
π/2	90°
π/3	60°
π/4	45°
π/6	30°

How to remember these fast:

Worked Example

a) Convert 43.8° to radians.

43.8° ÷ 180° = 73/300

(5π/4) ÷ π = 5/4

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Also in the full note

Topic 2 — Arcs & Sectors
What to Memorise
Concepts Checklist
Exam Tips
3What are arcs, minor arcs, and major arcs?
4Length of an arc
5Area of a sector

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Circular Measure