Question

How many sides do these regular Polygons have if their interior is 30?

Answer 1

`color(blue)("No such polygon")`

Explanation:

The formula for the sum of the interior angles of a regular polygon is given as:

`180^@n-360^@`

We do not know the number of sides, or the sum of the interior angles.

Lets call the sum of the angles S.

`180^@n-360^@=Scolor(white)(888)[1]`

Then one angle is `S/n`

`S/n=30color(white)(888)[2]`

Solving simultaneously:

From `[2]`

`S=30n`

Substituting in `[1]`

`180n-360=30n`

`150n=360=>n=360/150=12/5=2.4`

This is a fractional value, so it can't be the number of sides of a polygon. This means there is no polygon that has interior angles of `30^@`

We could have ascertained this at the beginning. As the number of sides of a polygon increase, the interior angles get larger.

A equilateral triangle is a regular polygon and has interior angles of `60^@`. A square has interior angles of `90^@`. You can see from this that to get an interior angle less than `60^@` you would need to have less than three sides. This is impossible.