Question

Each exterior angle of a regular polygon is approximately 21.18 degrees. What is the sum of the interior angles of a polygon?

Answer 1

Sum of interior angles `= color(green)(2700^@)`

Explanation:

If each exterior angle is `21.18^@`
then each interior angle is `180^@ - 21.18@=158.12^@`

For a regular polygon with `n` angles
the sum of the interior angles is `(n-1) * 180^@`

but since each of the `n` angles has an interior measure of `158.12`
the sum of the interior angles is also `n * 158.82`

So we have
`color(white)("XXX")(n-2) * 180 = 158.82 n`
which simplifies to
`color(white)("XXX")21.18n=360`
or
`color(white)("XXX")n=360/21.18~~17`

So the sum of the interior angles is
`color(white)("XXX")(17-2) * 180^@ = 2700^@`

Answer 2

The sum of the angles in a 17-sided polygon is 2700°

Explanation:

Sum of exterior angles in a polygon is always 360 degrees.

To find the number of side of polygons, we must divide 360 degrees by the size of the exterior angle

By using this property,

Number sides of polygon= `360/21.18`

Number of sides of the polygon = `16.997` (2 d.p.)
However, the number of sides must be a whole number.
There are 17 sides

After we get the number of sides of a polygon, we can find the total of the interior angle by using `180(n-2)`

Total of interior angles is `180xx(17-2)`= `2700°`