Sum of Interior Angles
Total of interior angles of an n-sided polygon.
Formula
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Example
Pentagon → 540°
How it works
For any polygon with n sides, you can split it into triangles by drawing diagonals from one vertex. This creates n − 2 triangles, and each triangle has 180° of angle sum, so the polygon’s interior angles add up to (n − 2) × 180°. This works for any simple polygon, whether regular or irregular, as long as it does not cross itself.
When to use it
Use this formula when you need the total of all interior angles in a polygon, such as in geometry problems or when checking missing angles in a shape. It is especially useful for finding one unknown angle after the others are known.
Worked example, step by step
Find the sum of the interior angles of a hexagon. Step 1: Identify the number of sides: n = 6 Step 2: Substitute into the formula: S = (n - 2) × 180° Step 3: S = (6 - 2) × 180° Step 4: S = 4 × 180° Step 5: 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°
- Counting the number of triangles incorrectly when splitting the polygon
- Thinking the formula gives one angle instead of the total of all interior angles
Did You Know?
The interior angles of any triangle always sum to 180° — but on a sphere they add up to more.
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