# What is the internal angle of a regular 17-sided polygon?

Dec 29, 2016

$\frac{15 \pi}{17}$ radians, or about ${158}^{\circ} 49 ' 24$"

#### Explanation:

A heptadecagon ($17$ sided polygon) can be dissected into $15$ triangles whose internal angles sum to the sum of the internal angles of the heptadecagon.

The internal angles of a (plane) triangle sum to $\pi$ radians or ${180}^{\circ}$.

So the total sum of the internal angles of a heptadecagon is:

$15 \pi$ radians $\text{ }$ or $\text{ } 15 \cdot {180}^{\circ} = {2700}^{\circ}$

So each interior angle in a regular heptadecagon is:

$\frac{15 \pi}{17}$ radians

${2700}^{\circ} / 17 \approx {158}^{\circ} 49 ' 24$"

Bonus

Note that $17$ is a Fermat prime since $17 = {2}^{{2}^{2}} - 1$ is of the form ${2}^{{2}^{n}} - 1$.

As a result, the regular heptadecagon is one of the few prime sided figures constructable using an unmarked ruler and pair of compasses - that is using a classical construction.

Dec 30, 2016

158.82°

#### Explanation:

A useful way to work with the angles in polygons is using the exterior angles.

The sum of the exterior angles of any polygon is always 360°

In a regular polygon with $n$ sides, the size of each exterior angle, $\beta$ can be calculated from:

beta = (360°)/n

The interior angle $\left(\theta\right)$of the polygon is the supplement of $\beta$

theta = 180°-beta

So, for a polygon with 17 sides,

theta = 180°- (360°)/17

theta = 180° -21.18°

theta = 158.82°