how do i figure these out???

Question

The apothegm of a regular pentagon is 10.8 centimeters.
what is the radius of the circumscribed circle?
what is the length of a side of the pentagon?
what is the perimeter of the pentagon?

Answer 1

Draw a regular pentagon; draw an apothem (a line segment from the center of the pentagon [the point where all the diagonals cross] perpendicular to a side of the pentagon; it also bisects that side). Draw a line segment from the center to one of the two vertices that are the endpoints of the side to which you drew the apothem. The latter line segment you have drawn is a radius of the circumscribed circle. This segment bisects the 108° angle at the vertex, so the angle opposite the apothem has measure 54°, and the angle formed by the apothem and the circumscribed circle radius is therefore 36°. The side of the triangle opposite this 36° angle is half the side of the pentagon. Denote the length of a side of the pentagon by S, the length of the apothem by A, and the length of the radius of the circumscribed circle by R. You know the value of A.

Then, from trig,

cos(36°) = A/R so R = A/cos(36°)
sin(36°) = (S/2)/R so S = 2R sin(36°)

where

cos(36°) = (1 + sqrt(5))/4 ≈ 0.8090
sin(36°) = sqrt(2(5 - sqrt(5))/4 ≈ 0.5878