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
What a learner can do afterwards
- 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
The lesson
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.
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°.
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°.
Split a polygon into triangles, multiply by 180°, then divide by n for a regular one's interior angle.
Watch it
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.