Mathematics (Core)Exam code: 0580

Circles, Arcs and Sectors

Lengths, Areas and Volumes

PractiseFlashcards

The vocabulary

  • An arc is part of the circumference, and a sector is the slice enclosed by two radii and an arc
  • The angle at the centre between the two radii is usually labelled θ

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 is kept to a factor of 360°, such as 90°, 60° or 45°, so the sector is always a simple fraction of the whole circle
  • 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

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