Polygons and Angle Sums
Interior angles of an n-sided polygon sum to (n − 2) × 180°.
About this topic
A polygon is a closed shape with straight sides. The interior angles of an n-sided polygon add up to (n − 2) × 180°, because it can be split into n − 2 triangles. The exterior angles of any convex polygon always add to 360°.
In a regular polygon every angle is equal, so each interior angle is (n − 2) × 180°/n. A hexagon has an angle sum of 720°, with 120° at each corner if regular, which is why hexagons fit together so neatly.
Worked example: a decagon has 10 sides, so its interior angles sum to (10 − 2) × 180° = 1440°, and each angle of a regular decagon is 144°. A regular polygon with each exterior angle 30° has 360/30 = 12 sides. If five angles of a pentagon are 100°, 110°, 120° and 90°, the last is 540 − 420 = 120°.
Designers use these sums for tiles, floors, stop signs and nuts and bolts. A typical mistake is using n × 180 instead of (n − 2) × 180, or forgetting that the exterior angle total is 360° no matter how many sides there are. Another is applying the regular formula to an irregular polygon, where the angles are not equal. Drawing diagonals from one vertex to split the shape into triangles helps you rebuild the formula whenever you forget it.
Key ideas
- Triangle 180°, quadrilateral 360°
- Exterior angles always sum to 360°
- Regular polygons have equal angles
Quick check
Use cases: where Polygons and Angle Sums helps
- Homework: review the key idea behind Polygons and Angle Sums before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Polygons and Angle Sums without notes.
- Visual learning: play with the interactive lab to see how Polygons and Angle Sums changes when you adjust its inputs.
- Teaching a friend: explain Polygons and Angle Sums in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Polygons and Angle Sums?
Interior angles of an n-sided polygon sum to (n − 2) × 180°.
What will I learn in Polygons and Angle Sums?
Triangle 180°, quadrilateral 360°. Exterior angles always sum to 360°. Regular polygons have equal angles.
Is Polygons and Angle Sums part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Polygons and Angle Sums lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Polygons and Angle Sums?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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