The perimeter of a regular pentagon is three more than seven times the approximate value of the apothem...?

Question

The area of the pentagon is 440. Find the length of a side and an approximation for the apothem.

Answer 1

Area = (1/2)*Apothem*Perimeter = 0.5*A*P

Given that
P = 7*A + 3

And

440 = 0.5*A*P...(I)

Substituting P in equation (l)

440 = 0.5*A*(7*A + 3)

880 = 7A² + 3A

7A² + 3A - 880 = 0

A = 11 or A = -80/7

Since A can't be negative A = 11

P = 7*11+3 = 77+3 = 80

Side = P/5 = 16

Answer 2

For an n-sided regular polygon, with side length s, apotherm a, we can write the area A as

A = nsa/2

hence 880 = 5sa

sa = 176

Now the perimeter for a pentagon is 5s, but here is also equal to 7a + 3

5s = 7a + 3

s = 0.2*(7a + 3)

Sub into the first expression

0.2*(7a + 3)*a = 176

1.4a^2 + 0.6a - 176 = 0

Simplify to get

7a^2 + 3a -880 = 0

Now you can solve this to get an expression for the apothem a = 11.

Work back to get s = 16

Answer 3

tan(36)=(.7a+.3)/a
.726a=.7a+.3
.026a=.3
a=11.3
side=(7*11.3+3)/5=16.42