Interior Angle Sum
Sum of interior angles of an n-sided polygon.
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Example
Pentagon: 540°
How it works
Any polygon can be split into triangles by drawing diagonals from one vertex. An n-sided polygon makes exactly n-2 triangles, and each triangle has 180° of interior angles, so the total interior-angle sum is (n-2)×180°. This works for simple polygons, whether they are regular or irregular.
When to use it
Use this when you need the total of all interior angles in a polygon, such as in geometry problems, angle-chasing, or checking if a shape is possible. It is also useful as a first step before finding individual angles in regular polygons.
Worked example, step by step
Find the sum of the interior angles of a hexagon. Use the formula: S = (n - 2) × 180° Substitute n = 6: S = (6 - 2) × 180° Simplify: S = 4 × 180° Calculate: S = 720° Final answer: The sum of the interior angles of a hexagon is 720°.
Common mistakes to avoid
- Using n × 180° instead of (n - 2) × 180°
- Forgetting that n means the number of sides, not the number of angles to subtract
- Using this formula for a shape that is not a simple polygon
Did You Know?
The interior angles of any triangle always sum to 180°, but only on a flat surface — on a sphere they add to more.
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