What is the perimeter of 5 equal sides and the area is 125?

Answer 1

The area of a regular polygon with n sides each sunit long, perimeter p, and apothem a is given by the formula

A = ans/2 or A = ap/2

The 5-sided polygon is a regular polygon (5 equal sides), so that its area equals

A = ans/2 (substitute 125 for A, and 5 for n)
125 = 5as/2 (multiply by 2 to both sides)
250 = 5as (divide by 5s to both sides)
50/s = a

Look at the right triangle formed by one half of the side s(base), the apothem a (height), and the side s(hypotenuse).
The angle opposite to the apothem equals 108⁰/2 = 54⁰ (one half of the interior angle of the pentagon). So we have:

tan 54⁰ = a/(s/2) (substitute 50/s for a)
tan 54⁰ = (50/s)/(s/2)
tan 54⁰ = 50/s x 2/s
tan 54⁰/1 = 100/s2 (cross multiply)
s2tan 54⁰ = 100
s2 = 100/tan 54⁰
s = √(100/tan 54⁰) (ignore the negative sign)
s ≈ 8.5

Thus, the side of the given pentagon is about 8.5 units long, and its perimeter is about 42.5 units.