For four gons, all angles are equal. Calculate how many sides does a polygon have with each angle 18

For four gons, all angles are equal. Calculate how many sides does a polygon have with each angle 18 degrees greater than each corner of the quadrilateral?

By the theorem on the sum of the angles of a quadrilateral: the sum of all interior angles of any quadrilateral is 360 °.
Since all the angles of the quadrilateral are equal to each other, we denote them as x. We get equality:

x + x + x + x = 360 °;

4 * x = 360 °;

x = 360 ° / 4;

x = 90 °.

According to the condition, it is necessary to calculate how many sides a polygon has, in which each angle is 18 ° larger than each corner of the quadrilateral, that is, you need to find the number of sides of a regular polygon in which each angle is 108 ° (90 ° + 18 °).
The degree measure of the angle of a regular polygon is found by the formula:

α = (180 ° * (n – 2)) / n,

where n is the number of sides of the polygon, α is the degree measure of the angle of the polygon.

Substitute the known values ​​and find the number of sides of the polygon:

(180 ° * (n – 2)) / n = 108 °;

108 ° * n = 180 ° * (n – 2);

108 ° * n = 180 ° * n – 180 ° * 2;

108 ° * n = 180 ° * n – 360 °;

108 ° * n – 180 ° * n = – 360 °;

– 72 ° * n = – 360 °;

n = (- 360 °) / (- 72 °);

n = 360 ° / 72 °;

n = 5.

Answer: 5 sides.