What it does
Why do the angles of a pentagon add up to 540°? Because any polygon can be cut into triangles, and each triangle’s angles add to 180°. This page lets you see that for any shape you like:
- Pick the number of sides, from 3 to 10, and drag the blue corners anywhere.
- The shape is cut into n − 2 triangles (shaded) and every corner angle is measured, then added up live next to the rule (n − 2) × 180°.
- Push a corner in makes a dent: one angle becomes a reflex angle (more than 180°), and the rule still works, because the shape still cuts into n − 2 triangles.
- Show exterior angles draws each side carried on past its corner and the turn to the next side. Those turns always add to 360°: one full turn as you walk around.
It fits Class 8 Mathematics, quadrilaterals (the angle sum of a quadrilateral and of polygons in general), and is useful for regular polygons and tilings later. For the special four-sided shapes, try the quadrilateral explorer; for drawing shapes with ruler and compass, interactive geometry construction.
How to use it
- Slide n to choose the number of sides. The shape starts regular.
- Drag any corner. The triangles, angles and sums follow your finger.
- Use Make it regular, Random shape or Push a corner in for quick shapes.
- Tick Show exterior angles to see the turning angles and their total.
A worked example
The page starts with a regular pentagon: 5 sides, so 5 − 2 = 3 triangles and 3 × 180° = 540°. Each angle is 540° ÷ 5 = 108°, and the angle line reads 108° + 108° + 108° + 108° + 108° = 540°. Turn on exterior angles: each is 180° − 108° = 72°, and 5 × 72° = 360°.
Now press Push a corner in. The top corner drops below the line joining its neighbours and its angle becomes more than 180°. The other angles shrink to make up for it, the sum stays 540°.
Check yourself
Five fresh questions on angle sums, regular polygons and exterior angles. Type a number, press Check, and open the worked answer if you need it.
FAQ
Is this from the NCERT book? It covers the same topic with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- The angle sum of a polygon, (n − 2) × 180°, and the exterior angle sum of 360° are standard Euclidean geometry. Wording, pictures and examples are fibub's own.
- Checked against hand-worked shapes (a square; an arrowhead with a 292.62° reflex corner, from 360° − 2 × atan(2/3)) and 500 random shapes, where the triangles must also cover exactly the shoelace area of the polygon.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-10.