help! precal!?

Question

The apothem of a regular pentagon is 10.8 centimeters.

a) Find the radius of the circumscribed circle.
b) What is the length of a side of the pentagon?
c) Find the perimeter of the pentagon.

Answer 1

The apothem of a regular pentagon is 10.8 centimeters.

a) Find the radius of the circumscribed circle.
b) What is the length of a side of the pentagon?
c) Find the perimeter of the pentagon.
__________

Let
a = apothem
r = radius of circumscribed circle.
s = length of side of pentagon
_________

a) Find the radius of the circumscribed circle.

The central angle that subtends one side of the pentagon is:
360/5 = 72°. You can bisect that angle and create a right triangle such that:

cos(72°/2) = a/r
r = a/cos(36°) = 10.8/cos(36°) ≈ 13.349534 cm
_________

b) What is the length of a side of the pentagon?

From the same right triangle in a):

tan(72°/2) = (s/2)/a = s/(2a)
s = 2a*tan(36°) = (2*10.8) tan(36°) ≈ 15.693319 cm
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c) Find the perimeter of the pentagon.

Perimeter = 5s = 5*(2*10.8) tan(36°) ≈ 78.466593 cm