Interior Angle (Regular Polygon)
Measure of each interior angle of a regular polygon.
Formula
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Example
A regular pentagon (n=5): each angle = 3·180°/5 = 108°.
How it works
A regular polygon has all sides and all angles equal, so once you know the number of sides, you can find the size of every interior angle. The formula works because a shape with n sides can be split into (n-2) triangles from one vertex, and the angles in those triangles add up to (n-2) × 180°. Dividing that total equally among the n equal interior angles gives the size of one angle.
When to use it
Use this when you need the size of each angle in a regular shape such as an equilateral triangle, square, regular hexagon, or any other regular polygon. It is useful in geometry problems, design, tiling, architecture, and checking whether a polygon can fit evenly around a point.
Worked example, step by step
Find each interior angle of a regular octagon. Step 1: Identify the number of sides: n = 8 Step 2: Substitute into the formula: θ = ((n - 2) × 180°) / n Step 3: θ = ((8 - 2) × 180°) / 8 Step 4: θ = (6 × 180°) / 8 Step 5: θ = 1080° / 8 Step 6: θ = 135° Final answer: each interior angle of a regular octagon is 135°.
Common mistakes to avoid
- Using the formula for the sum of interior angles instead of the size of one angle
- Forgetting to subtract 2 from the number of sides before multiplying by 180°
- Applying this formula to irregular polygons, where the angles are not all equal
Did You Know?
A regular hexagon's 120° angles are why honeybees build hexagonal cells — they tile with no wasted space.
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