How many sides does a convex polygon have if each angle is 135 degrees?

By hypothesis, all angles of a convex polygon are equal to 135 °, then this polygon is regular.

The degree measure of the angle of a regular polygon is calculated by the formula:

α = (n – 2) / n * 180 °,

where α is the angle of a regular polygon, n is the number of sides of a regular polygon.

Let’s substitute the data on the value condition into the formula:

(180 ° * (n – 2)) / n = 135 °;

180 ° * (n – 2) = 135 ° * n (in proportion);

180 ° * n – 180 ° * 2 = 135 ° * n;

180 ° * n – 135 ° * n = 180 ° * 2;

45 ° * n = 360 °;

n = 360 ° / 45 °;

n = 8.

Answer: A convex polygon with each angle of 135 ° has 8 sides.