---
title: "Angle sums in triangles and polygons"
description: "Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons"
canonical: https://lightmysky.com/learn/mathematics/angle-sums-in-triangles-and-polygons-mt_J03RFlVdas
source: https://lightmysky.com/learn/mathematics/angle-sums-in-triangles-and-polygons-mt_J03RFlVdas.md
retrieved: 2026-10-19
---

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# Angle sums in triangles and polygons

Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons

Subject: Mathematics · Area: Geometry · Ages 11 to 14
Page: https://lightmysky.com/learn/mathematics/angle-sums-in-triangles-and-polygons-mt_J03RFlVdas

## Ready when they can

- Calculate a missing angle in a triangle by subtracting the known angles from 180°
- Find the sum of interior angles in a hexagon by dividing it into triangles
- Calculate each interior and exterior angle of a regular polygon given the number of sides

## Lesson: Angle sums in triangles and polygons

You already know that the three angles inside any triangle always add up to 180°. That one fact is the key to every other shape too.

*(drawing: Draw lines from one corner of a hexagon to the other corners. It splits into 4 triangles.)*

**Example.** Try a pentagon, which has 5 sides. Draw lines from one corner and it splits into 3 triangles. Three triangles means 3 times 180°, which is 540°. That is the angle sum for any pentagon.

For any polygon with n sides, the number of triangles from one corner is always (n − 2). So the angle sum is (n − 2) × 180°.

**Tip.** In a regular polygon every interior angle is equal, so divide the total by n. Each exterior angle is 180° minus the interior angle, and all the exterior angles of any polygon always add up to 360°.

Two more facts help with regular polygons. In a regular polygon every interior angle is equal, so each one is the total angle sum divided by n. Each exterior angle sits on a straight line with its interior angle, so one exterior angle is 180° minus the interior angle. And if you walk all the way around any polygon, turning one exterior angle at each corner, you make exactly one full turn: the exterior angles of any polygon always add up to 360°, however many sides it has.

**Recap.** Split a polygon into triangles, multiply by 180°, then divide by n for a regular one's interior angle.

## Practice

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

## Needs first

- [Angles in triangles (age 10+)](https://lightmysky.com/learn/mathematics/angles-in-triangles-age-10-mt_cdMlC7EpTJ)
- [Algebraic Notation](https://lightmysky.com/learn/mathematics/algebraic-notation-mt_KwdjWEmMNo)
- [Understanding angles (age 11+)](https://lightmysky.com/learn/mathematics/understanding-angles-age-11-mt_tedML_iu4Y)