Question

What is the formula for finding exterior and interior angles of a polygon?

Answer 1

Each interior angle of a regular polygon with `n` sides:

`color(red)(theta = (180(n-2))/n" or "theta = (180n-360)/n`

Each exterior angle of a regular polygon with `n` sides:

`color(green)(beta = 180°-theta`

Note that interior angle + exterior angle = `180°`

`theta = 180°-beta " and " beta = 180°-theta`

Explanation:

To find the size of each interior angle of a regular polygon you need to find the sum of the interior angles first.

If the number of sides is `n`, then

the sum of the interior angles is:

`color(blue)(S = 180(n-2))`

This formula derives from the fact that if you draw diagonals from one vertex in the polygon, the number of triangles formed will be `2` less than the number of sides. Each triangle has `180°`.

The formula can also be used as `color(blue)(S = 180n-360)`

This form of the formula derives from drawing triangles in the polygon by drawing lines from a central point to each vertex. In this way the number of triangles is the same as the number of sides, but the angles at the centre are not required, so `360°` is subtracted.

Once you have the sum of all the interior angles you divide by the number of sides to find

the size of each interior angle

`color(red)(theta = (180(n-2))/n)" or " color(red)(theta = (180n-360)/n)`

To find the size of each exterior angle, `beta`, subtract `theta` from `180°`

`color(green)(beta = 180°-theta`

Another method to find the exterior angle is using the fact that the sum of the exterior angles is always `360°`

`color(green)(beta = (360°)/n`

Once you know the size of the exterior angle you can find the size of the interior angle by subtracting from `180°`

`color(red)(theta = 180°-beta)`