Roman C. answered • 03/09/13

Masters of Education Graduate with Mathematics Expertise

A few notes:

As the other tutors said, more information is needed to find the exact area as different shapes have different areas.

However there are nice facts that can be shown about the area given the perimeter.

1. If the shape is apolygon with n sides, the maximum area possible occurs if the polygon is regular.

For example in your question, if n = 4 (quadrilateral), the maximum area occurs if it is a square with 4 in. on a side yielding 16 sq. in. for the area.

2. If any shape is allowed, even ones with curved sides, then the largest area is that of a circle.

Thus we can get the maximum area achievable by using P = 2πr for the perimeter (in this case circumference) and A = πr^{2}.

We find that r = P/(2π) so A = π(P/(2π))^{2} = P^{2}/(4π). Any positive area less than this is also possible.

So in this problem the largest area possible is (16 in.)^{2}/(4π) = 64/π sq. in. ≈ 20.37 sq. in.