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Plane Figures

Interior and Exterior Angle Measures of Polygons

Interior and Exterior Angles of Polygons

Since a pentagon can be divided into triangles and a hexagon can be divided into triangles, the sum of the interior angle measures for each is:
In general, for an -sided polygon, drawing all diagonals from a single vertex divides the polygon into triangles. Thus, the sum of the interior angle measures of an -sided polygon is:

Sum of Interior Angles of an -Sided Polygon

The sum of the interior angles of an -sided polygon is given by:

For a regular polygon, all interior angles are equal. Therefore, the measure of each interior angle in a regular -sided polygon can be calculated by dividing the sum of the interior angles by the number of vertices:
(Measure of each interior angle of a regular -sided polygon)

Sum of Exterior Angles of a Polygon

In any polygon, the sum of the measures of an interior angle and its corresponding exterior angle at any vertex is always . Since there are vertices in an -sided polygon:
The sum of the interior angle measures of an -sided polygon is . Therefore, the sum of the exterior angle measures is:

Sum of Exterior Angles of an -Sided Polygon

The sum of the exterior angles of an -sided polygon is always .

For a regular polygon, all exterior angles are equal. Thus, the measure of each exterior angle in a regular -sided polygon can be calculated by dividing the sum of the exterior angles by the number of vertices:
(Measure of each exterior angle of a regular -sided polygon)
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