The area of a polygon is 48 mi2. What is the formula that shows the length of side a?

Answer 1

unless it is a regular polygon, there is no such formula.
For a regular polygon of n sides and area A, the side length is

a = √(A/n tan(π/n))

so, once you have figured out how many sides you have, apply the formula.

Answer 2

I assume your polygon is a regular polygon, that is,
all sides are equal, and all interior angles are equal.

Let the number of sides be n and consider one of the isosceles
triangles consisting of a base of a , two equal sides and a central angle
of 360/n°
construct a perpendicular so you have a right-angled triangle
with base a/2 , a height of h, and the angle at the centre as 360/(2n)°
tan (360/(2n)) = a/2 / h
h = (a/2) / tan(180/n)°

area of one triangle = (1/2) base x height
= (1/2)(a/2)tan(180/n) = (1/4) a tan(180/n)
you have n of those, so total area =
(n/4)a tan(180/n) = 48
a n tan(180/n) = 192
a = 192/(ntan(180/n)

so once you know what n is in your polygon, you got it.

Answer 3

Oops. I forgot the factor of 2.
a = 1/2 √(A/n tan(π/n))

I think mathhelper forgot a factor of a. Area is in units^2, not just units.

Answer 4

yup, glad you caught that.

Would this work?

suppose we have a hexagon with side of 6, then each of the 6 triangles has an area of
(1/2)(6)(3√3) = 9√3
total area = 54√3 = appr 93.53

ratio of areas of similar shapes is proportional to the square of their sides.
93.53/48 = 36/a^2
a^2 = 48*36/93.53 = 18.47..
a = appr 4.3