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Interior and exterior angles

Angles & polygons

Japanese school year: Junior high 2

What you learn

Learn how to calculate the sum of interior angles by dividing polygons into triangles, and understand why exterior angles always sum to 360 degrees. This is useful for analyzing regular polygons and architectural shapes. Prior knowledge of triangle angle sums being 180 degrees is required.

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Key points

This formula is used to calculate the sum of all interior angles SS in an nn-sided polygon. Because the polygon can be divided into (n−2)(n - 2) triangles from a single vertex, multiply 180∘180^\circ by (n−2)(n - 2).

S=180∘×(n−2)S = 180^\circ \times (n - 2)

This formula represents the sum of all exterior angles θext\theta_{\text{ext}} of a polygon. No matter how many sides the polygon has, its exterior angles always add up to 360∘360^\circ.

∑θext=360∘\sum \theta_{\text{ext}} = 360^\circ

This formula is used to find the size of one interior angle θ\theta in a regular nn-sided polygon. Divide the sum of all interior angles by the number of corners nn.

θ=180∘×(n−2)n\theta = \frac{180^\circ \times (n - 2)}{n}

This formula is used to find the size of one exterior angle φ\varphi in a regular nn-sided polygon. Divide the total exterior angle sum (360∘360^\circ) evenly by the number of vertices nn.

φ=360∘n\varphi = \frac{360^\circ}{n}

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