Angles and PolygonsAngles in Polygons
Every polygon has interior angles — the angles formed inside the polygon at each vertex. Let's discover a pattern!
Move the vertices of this triangle and watch how the three interior angles change:
No matter how you move the vertices, the three angles always add up to
Angle Sum of a Triangle
The interior angles of any triangle always add up to 180°.
We can use this to find the angle sum of larger polygons. A
Since each triangle has an angle sum of 180°, the quadrilateral has an angle sum of
We can do the same for any polygon — split it into triangles from one vertex:
| Polygon | Sides | Triangles | Angle Sum |
|---|---|---|---|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| n-gon | n | n − 2 | (n − 2) × 180° |
Interior Angle Sum Formula
The sum of the interior angles of an n-sided polygon is
For a
Use the slider below to see how the interior angle changes as the number of sides increases:
A regular triangle has angles of