Sum of Interior Angles of a Polygon
Main Concept
An interior angle is an angle located inside a shape. For example, a triangle has three interior angles, with a sum of 180°. As the number of sides, n, of a polygon increases, the sum of its interior angles, S, also increases by the following formula:
S = n−2⋅180°
Any polygon with n sides can be split into a union of n−2 triangles. This follows mathematical induction from two properties:
First property:
Base case: The sum of interior angles of a triangle is 180°.
Induction step: If a side is added, the sum of the angles must increase by 180°.
Second property:
If a side is added, it is equivalent to the combination of two operations:
Splitting any existing side into two sides is possible by declaring a new vertex at a point anywhere along that side. The angle at the new vertex is 180°. As a result, the sum of the angles increases by 180°.
Adjusting the angles at all the vertices to get the new shape without changing the number of sides is possible. If one angle is increased, another angle must be decreased by the same value, otherwise the polygon will no longer be closed. The sum of the angles does not change using this operation.
In general, with regular polygons (all sides and angles are equal), each angle is equal to:
A = n−2⋅180°n
Adjust the slider to change the number of sides (n) on the polygon below.
Click "View Triangles" to see each angle of the polygon, as well as the sum of its interior angles.
Number of Sides:
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