0580

Circles, Arcs and Sectors

Lengths, Areas and Volumes

The vocabulary

  • An arc is part of the circumference, and a sector is the slice enclosed by two radii and an arc
  • Two points on a circle cut it into a shorter minor arc and a longer major arc, and two radii likewise cut it into a minor and a major sector
  • The angle at the centre between the two radii is usually labelled θ
  • A question naming the major arc or sector wants the larger piece, so subtract the marked angle from 360° before doing anything else

The two formulas

  • An arc and a sector are simply fractions of the whole circle, and the fraction is θ ÷ 360
  • Arc length = (θ ÷ 360) × 2πr
  • Sector area = (θ ÷ 360) × πr²
  • Neither formula is printed in the paper, so both have to be known, although each is just the circle formula scaled by the fraction
A sector drawn with radius r and the angle theta marked at its point, beside the two formulas it needs: sector area equals theta over 360 times pi r squared, and arc length equals theta over 360 times 2 pi r
Source: Arc Lengths & Sector Areas by Save My Exams
  • θ can be any angle, and the syllabus explicitly includes major sectors, where θ is bigger than 180°
  • To go all the way round a sector you travel along the arc and back down both radii, so its perimeter includes those two straight edges — routinely forgotten
Worked example

Arc, area and perimeter of a sector

A sector of a circle has radius 12 cm and sector angle 45°. Find its arc length, its area, and its perimeter. Give the first two in terms of π.

A sector of a circle with centre O marked at its point. The two straight radii are drawn, one of them labelled 12 cm, and the angle between them at O is marked 45 degrees. Neither the arc length nor the area is shown.

Solution:

  • The fraction of the circle is 45 ÷ 360 = 1/8
  • The full circumference is 2π × 12 = 24π cm
  • Arc length = 1/8 × 24π = 3π cm
  • The full area is π × 12² = 144π cm²
  • Sector area = 1/8 × 144π = 18π cm²
  • The perimeter is the arc plus both radii: 3π + 12 + 12
  • Perimeter = (3π + 24) cm
Worked example

A major sector

A sector of a circle, centre O, has radius 10 cm. The minor sector angle is 72°. Find the area of the major sector and the length of the major arc, in terms of π.

A circle with centre O and two radii drawn, one labelled 10 cm. The angle between them is marked 72 degrees and the smaller wedge it cuts off is left white, while the whole of the rest of the circle, the major sector, is shaded grey. The 288 degree angle is not marked.

Solution:

  • The major sector is the rest of the circle, so its angle is 360 − 72 = 288°
  • The fraction is 288 ÷ 360 = 4/5
  • The full area is π × 10² = 100π cm²
  • Major sector area = 4/5 × 100π = 80π cm²
  • The full circumference is 2π × 10 = 20π cm
  • Major arc = 4/5 × 20π = 16π cm
  • Check: the minor sector would be 1/5 of the circle, and 80π + 20π = 100π