Question

What is the area of a square with the same perimeter as a regular hexagon with area 24 square root of 3.

Can you please help me wodrk this?

Answer 1

area of hexagon with side s is 3√3/2 s^2
so, if

3√3/2 s^2 = 24√3
s^2 = 24√3 * 2/3√3 = 16
s = 4
p = 4*6 = 24

for the square, if p=24, s=6
area = 36

Note that 24√3 = 41.57 > 36

The area of the hexagon is greater than that of the square. As the number of sides increases, the area increases, until the maximum area for a given perimeter is given by a circle.

Answer 2

area of hexagon = 24√3
so area of one of the equilateral triangles of the hexagon = 4√3
It can be shown easily that the area of an equilateral triangle with sides 4 is 4√3
so the side of the hexagon is 4
and its perimeter would be 24

So each side of the square is 6
and the area of the square is 36