--- 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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 A pizza cut from the centre by equal angles gives equal slices. An arc is a fraction of the circle's edge. Divide the central angle by 360 for the fraction, then multiply by the circumference, 2 pi r. The radius is half the diameter, so circumference is also pi times diameter. A sector is bounded by two radii and an arc. A segment by a chord and an arc. Keep pi as a symbol when asked, or use 3.14. A 90 degree angle gives a quarter. To find radius from an arc and nonzero angle, multiply arc by 360 over the angle, then divide by 2 pi. *(drawing: A 90 degree central angle gives a quarter of the circle.)* **Example.** A sector is the same fraction of the area. With radius 8 cm and a central angle of 45 degrees, the fraction is one eighth, so the area is one eighth of 64 pi, which is 8 pi cm squared. With pi as 3.14, that is 25.12 cm squared. A minor segment is what is left when you cut the triangle out of its sector, for a central angle between 0 and 180 degrees. Work out the sector, work out the triangle, then subtract. The triangle has two sides equal to the radius and the central angle between them, so its area is one half times r squared times sin of the angle. For a 90 degree angle this is one half r squared, the right-angled case. At 180 degrees the triangle vanishes and the segment equals a semicircle. For a major segment, with the angle between 180 and 360 degrees, the segment is the sector plus the triangle, since the triangle area is positive. 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. For a minor segment, with the angle between 0 and 180 degrees, subtract the triangle. For a major segment, with the angle between 180 and 360 degrees, add it. ## 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)