what is the side of a regular pentagon in a circle with a circumference of 770 inches

Question

what is the side of a regular pentagon in a circle with a circumference of 770 inches

Answer 1

he radius of the circle to each vertex of the pentagon forms

5 congruent isoceles triangles. Inside each triangle, the center angle

is 360/5 = 72 degrees. This force the base angles to be 54 degrees each.

Dropping the height from the center of the circle to the base of each

triangle splits each one into two smaller right triangles with

acute angles of 36 at the center and 54 at the base. Specifically,

the height splits the 72 degree angle into 2 x 36, by HL theorem.

Since the circumference is 770, the diameter is 770/pi. The radius is

half of that which is 122.549....

cos(54) = x / 122.549....

x = cos(54)* 122.549 = 72.0324

The side of each pentagon is then 2x = 144.06....