# The sum of the measures of the interior angles of a polygon is 720°. What type of polygon is it?

Jul 18, 2018

Hexagon

#### Explanation:

Let's do this two ways: Using a pattern we know and using exterior angles.

Pattern
For a triangle, the sum of interior angles is 180 degrees.
For a quadrilateral, the sum of interior angles is 360 degrees.

We know that there is some pattern, so we can deduce quite logically that a pentagon has a sum of 540 degrees and a hexagon has a sum of 720 degrees, hence the answer is a hexagon.

Exterior Angles
The real question is how do we find that there is a pattern? We can recall that the sum of all exterior angles in any polygon is 360 degrees. If you don't remember learning this, you can imagine driving along the perimeter of any polygon and seeing that, in the end, you've taken a single turn.

Anyway, in a normal polygon with $n$ sides, each of these exterior angles is, therefore, ${360}^{\setminus} \frac{\circ}{n}$. Since an external angle plus an interior angle is a line, we know that an interior angle will have measure ${180}^{\setminus} \circ - {360}^{\circ} / n$.

With $n$ of those angles, we get the total internal angle of
${\left(180 n - 360\right)}^{\circ}$. Setting this equal to ${720}^{\circ}$, we easily find $n = 6$.

Hexagon

#### Explanation:

A polygon of $n$ number of sides can be divided into $n - 2$ number of triangles.

Let the polygon have $n$ number of sides then the sum of its interior angles will be equal to the sum of interior angles of $n - 2$ triangles

$= \left(n - 2\right) {180}^{\setminus} \circ$

${720}^{\setminus} \circ = \left(n - 2\right) {180}^{\setminus} \circ$

$n - 2 = \frac{720}{180}$

$n - 2 = 4$

$n = 6$

Hence the polygon has $6$ sides hence it's a hexagon