U15 Angles in Polygons: Essential Formulas

This section covers the key formulas and theorems related to interior and exterior angles of polygons, which are fundamental for solving DSE geometry problems.

1 Sum of Interior Angles

Formula for Sum of Interior Angles

For any convex polygon with $n$ sides, the sum of its interior angles is given by the following formula.

$$ S = (n - 2) \times 180^\circ $$
n-sided polygonn = number of sides

2 Sum of Exterior Angles

Theorem for Sum of Exterior Angles

For any convex polygon, the sum of its exterior angles (one at each vertex) is always constant, regardless of the number of sides $n$.

$$ \sum \text{Exterior Angles} = 360^\circ $$
Sum of all red exterior angles = 360°

3 Interior and Exterior Angle of a Regular Polygon

Formulas for Regular Polygons

For a regular polygon (all sides and angles equal) with $n$ sides, each interior and exterior angle can be calculated as follows.

$$ \text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n} $$

or equivalently,

$$ \text{Each interior angle} = 180^\circ - \text{Each exterior angle} $$
$$ \text{Each exterior angle} = \frac{360^\circ}{n} $$
Regular Hexagon (n=6)Each int. angle = 120°Each ext. angle = 60°

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