Circles, arcs and sectors

• Circumference C = 2πr and area A = πr² (both given on the List of formulas).
• Work backwards from a circumference or area to the radius.
• Arc length = (θ ÷ 360) × 2πr; sector area = (θ ÷ 360) × πr².
• Major arcs and sectors use the reflex angle 360° − θ.
• The perimeter of a sector includes the two radii.
• Answers may be asked for in terms of π, which is how Paper 1 (no calculator) usually asks.

C = 2πr, A = πr²

Both formulas use the radius. If you are given the diameter, halve it first (or use C = πd). Use the π button or 3.142. "In terms of π" means leave π in the answer, e.g. 25π cm².

From area or circumference to radius

Rearrange the formula: r = C ÷ 2π, or r = √(A ÷ π).

Remember the straight edges

A semicircle has half the area of the circle. Its perimeter is half the circumference plus the diameter. A quarter circle has a quarter of the circumference plus two radii.

The fraction θ/360

A sector with angle θ at the centre is θ/360 of the whole circle, so its arc is θ/360 of the circumference and its area is θ/360 of the area. These two are not printed on the List of formulas: build them from C = 2πr and A = πr², which are.

Reflex angles and rearranging

The major sector uses the reflex angle 360° − θ. To find an angle, put the known arc or area over the whole circumference or area and multiply by 360.