Arc Length and Sector Area

Unit 10: Area of Polygons and Circles · ~15–30 min

A fraction of the whole circle

An arc is a piece of a circle's edge, and a sector is the pie-slice region it borders. Both are controlled by the same idea: a central angle of measure θ (in degrees) marks off θ/360 of the full circle. Multiply that fraction by the whole-circle formula — circumference for arc length, area for sector area — and you get the piece.

• Arc length: L = (θ/360) × 2πr.
• Sector area: A = (θ/360) × πr².

Notice both formulas are just "fraction of the circle" times "whole-circle amount." A full circle (θ = 360°) gives back the original circumference or area exactly, which is a quick way to sanity-check either formula.

Guided practice

• 1.Find the arc length of a 45° central angle in a circle with radius 16 m, in terms of π.
  Show answer

  L = (45/360)(2π)(16) = (1/8)(32π) = 4π m.
• 2.Find the sector area for a 120° central angle in a circle with radius 9 ft, in terms of π.
  Show answer

  A = (120/360)π(9)² = (1/3)(81π) = 27π ft².
• 3.A sector has area 15π cm² in a circle with radius 6 cm. Find the central angle.
  Show answer

  15π = (θ/360)π(36) = 0.1θπ, so θ = 150°.
• 4.Explain why arc length and sector area both use the same fraction θ/360, even though one measures length and the other area.
  Show answer

  The central angle θ determines what portion of the full 360° circle the slice covers, and that same proportion applies uniformly to any quantity tied to the whole circle — whether it's the circumference being measured (arc length) or the area being measured (sector area).
• 5.A circle has circumference 40π in. A sector has arc length 5π in. Find the sector's central angle.
  Show answer

  5π = (θ/360)(40π), so θ/360 = 5/40 = 1/8, giving θ = 45°.