# What is the measure of one interior angle of a regular 15 sided polygon?

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22
Mar 5, 2018

156°

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Number the vertices with consecutive integers 1 through 15. Let the edge (side) of the polygon connecting vertex i and vertex i+1 be called edge $i$, for all i ($1 \le i \le 14$), and let the edge connecting vertices $15 \mathmr{and} 1$ be called $15$. Then, if you imagine walking clockwise around the periphery of the polygon, from vertex 1 back to vertex 1, you will turn to the right $15$ times. The angles through which you have turned are the exterior angles of the polygon.

Since the polygon is regular, all sides and angles are equal, so each turn at each vertex is the same, and of size 360°/15 = 24° degrees. That is, since you've returned to the starting point you must have completely turned around, i.e. turned 360°. Now, if you turn 15° to the right at each vertex, the amount you didn't turn, i.e. the size of the interior angle, is 180-24=156°.

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(2340°)/15 = 156°

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In any convex polygon, if you start at one vertex and draw the diagonals to all the other vertices, you will form triangles,

The number of triangles so formed is always $2$ LESS than the number of sides. As each triangle has 180°, you can find the sum of the interior angles of the polygon:

For an $n$-sided polygon there are $\left(n - 2\right)$ triangles.
The sum of the interior angles is therefore 180°(n-2)

In a $15$-sided polygon:

Sum interior angles = 180(15-2) = 180 xx13 = 2340°

Each interior angle of the regular polygon = (2340°)/15 = 156°

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