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Angles in Polygons

Year 8 📐 Geometry & Measures Interior and exterior angles of polygons, angle sum rules.

The interior angles of any polygon sum to a value that depends on the number of sides $n$.

⚡ Interior Angle Sum

$$S = (n - 2) \times 180°$$

Example: Pentagon ($n=5$): $S = 3 \times 180 = 540°$

💡 You can derive this by splitting any polygon into triangles from one vertex — a polygon with $n$ sides splits into $n-2$ triangles.

In a regular polygon, all sides are equal and all angles are equal.

⚡ Each Interior Angle (regular)

$$\text{Interior angle} = \frac{(n-2) \times 180°}{n}$$

⚡ Each Exterior Angle (regular)

$$\text{Exterior angle} = \frac{360°}{n}$$

Example: Regular hexagon: exterior angle $= 60°$, interior angle $= 120°$

The exterior angles of any polygon (one at each vertex, measured going around once) always add up to 360°.

💡 Interior angle + exterior angle = 180° (they form a straight line). Use this to find one from the other.

Example: A regular polygon has exterior angle 40°. Number of sides $= 360 \div 40 = 9$ — it is a nonagon.

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