---
title: "Arc Length, Sector Area and Segments"
description: "Find arc lengths and sector areas as fractions of a whole circle, and find a segment by subtracting a triangle from its sector."
canonical: https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ
source: https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ.md
retrieved: 2026-09-12
---

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# Arc Length, Sector Area and Segments

Find arc lengths and sector areas as fractions of a whole circle, and find a segment by subtracting a triangle from its sector.

Subject: Mathematics · Area: Geometry · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ

## Ready when they can

- Work out an arc length as the angle over 360 of the circumference
- Find a sector area and leave it in terms of pi when asked
- Find a segment area by subtracting the triangle from the sector

## Lesson: Arcs, sectors and segments

Sharing a round pizza fairly means cutting by angle, so each slice is the same fraction of the whole. An arc is that fraction measured along the circle's edge. Divide the angle by 360 to get the fraction, then multiply by the circumference. Keep pi as a symbol when asked, or use 3.14 for a decimal. A 90 degree angle always gives a quarter of the circle.

**Example.** A sector is the same fraction of the area. With radius 6 cm and angle 90 degrees, the fraction is a quarter, so the area is a quarter of 36 pi, which is 9 pi. With radius 10 cm and angle 72 degrees, the fraction is a fifth, so with pi as 3.14 the area is 62.8.

A segment is what is left when you cut the triangle out of its sector. Work out the sector, work out the triangle, then subtract. With radius 8 cm and angle 90 degrees, the sector is 50.24 and the triangle is 32, so the segment is 18.24.

A 180 degree sector is exactly half the circle, for arc length and area alike. A 360 degree sector is the whole circle. Use these two facts to sense check every answer.

**Recap.** Scale the full circumference or area by angle over 360, then subtract the triangle for a segment.

## Practice

20 questions on this page, each with its working shown.

## Needs first

- [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn)
- [Triangle Area with Half ab sin C](https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b)
- [Circles: Circumference & Area](https://lightmysky.com/learn/mathematics/circles-circumference-and-area-mt_svFa6_mjO_)

## Opens up

- [Circle Theorems: Angles at the Centre and in a Semicircle](https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR)
- [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4)