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Maths

Circles: Arcs, Sectors and Segment Area

Overview

Students extend their knowledge of circle area and circumference to sectors and arcs — parts of circles defined by an angle at the centre — and apply these to practical problems.

Learning Objective
Students calculate the arc length and area of a sector using angle as a fraction of the whole circle, and find the area of a segment using sector area minus triangle area.

Resources needed

  • Calculator
  • Compass and protractor (optional)
  • Mini whiteboards

Lesson stages

0 / 7 done
  1. 1 Area = πr². Circumference = 2πr = πd. Quick calculations: area of circle with r=5cm, circumference with d=10cm. Establish these are the '100%' or 'whole circle' values.
  2. 2 A sector is a 'slice of pie' — the angle at the centre tells us what fraction of the whole circle it is. A 90° sector = 90/360 = 1/4 of the circle. Arc length of 90° sector, r=6: 1/4 × 2π×6 = 3π ≈ 9.42cm.
  3. 3 Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². Students apply these to four sectors with different angles and radii. Include non-standard angles (e.g. 135°, 72°).
  4. 4 The perimeter of a sector includes the arc AND two radii. Total perimeter = arc length + 2r. Calculate for θ=60°, r=9cm: arc = (60/360)×2π×9 = 3π ≈ 9.42. Perimeter = 9.42 + 18 ≈ 27.4cm. Students find three perimeters.
  5. 5 A segment is the region between a chord and the arc. Area = sector area − triangle area. For θ=60°, r=10: sector area = (60/360)×π×100 = 100π/6. Triangle (isosceles, two sides = 10, included angle 60° — equilateral!): area = ½ × 10 × 10 × sin60°. Subtract.
  6. 6 A sector has arc length 12cm and radius 8cm. Find the angle. Arc = (θ/360)×2π×8. 12 = (θ/360)×16π. θ = (12×360)/(16π) ≈ 85.9°. Students solve two reverse problems.
  7. 7 A sprinkler covers a 120° arc of radius 4m. What area of lawn does it water? A pizza slice has radius 15cm and angle 40°. What is the area of the crust (outer arc region, 1cm wide)?

Tap a step to mark it as done.

Variations

  • Express answers in terms of π (exact form)
  • Extend to 3D: surface area of a cone using sector
  • Find the radius given sector area and angle
More information

Display: sector, arc, segment, chord, radius, diameter, circumference. Teach: 'arc is the curved edge; sector is the slice shape; segment is the region between chord and arc.'

Provide formula cards. Restrict to quarter and third sectors (90° and 120°) before general angles. Focus on one formula at a time.

Do students use the angle as a fraction of 360 (not of 180 or 100)? Do they include two radii in the perimeter? Do they subtract triangle area for segments?

Calculator required for π. All other work on mini whiteboards or plain paper.

Students may use the wrong fraction (e.g. divide by 180 for a semicircle). Reinforce: the whole circle has 360°, always divide by 360.

Activity details
Secondary Challenging 45 minutes 5 minutes prep Solo, Pairs Indoor
Topics
Circles Arcs Sectors Area Geometry