# What do the interior angles of a polygon have to add up to?

Nov 24, 2016

See below.

#### Explanation:

The sum of interior angles of a polygon is equal to

$A = \left(\frac{\pi}{2}\right) n$

where $n$ is the number of triangles needed to compose the polygon.

Occours that the number of triangles is equal to the number of sides minus $2$ or

$n = l - 2$ where $l$ is the number of polygon sides.

The final formula is

$A = \left(\frac{\pi}{2}\right) \left(l - 2\right)$

Examples.

A triangle has $l = 3$ so $A = \frac{\pi}{2}$
A quadrilateral has $l = 4$ so $A = \pi$
A heptagon has $l = 7$ so $A = 5 \left(\frac{\pi}{2}\right)$

Nov 24, 2016

To determine the sum of the internal angles of a polygon, take the number of sides on the polygon, subtract 2 and multiply by 180 degrees.

#### Explanation:

A triangle has three sides. $\left(3 - 2\right) \cdot 180 = 180$
The sum of the internal angles of a triangle is 180 degrees.

A quad lateral has four sides. $\left(4 - 2\right) \cdot 180 = 360$
The sum of the internal angles of a square is 360 degrees.

A pentagon has 5 sides. $\left(5 - 2\right) \cdot 180 = 540$
The sum of the internal angles of a pentagon is 540 degrees.

A hexagon has six sides. $\left(6 - 2\right) \cdot 180 = 720$
The sum of the internal angles of a hexagon is 720 degrees

A heptagon has seven sides. $\left(7 - 2\right) \cdot 180 = 900$
The sum of the internal angles of a heptagon is 900 degrees.

and so on!