Suppose that S is a planar region enclosed by a square having an edge of length

Question

Suppose that S is a planar region enclosed by a square having an
edge of length L, that C is a planar region enlosed by a circle having
radius r, and that H is a planar region enclosed by a regular hexagon
having an edge of length L. In each case, compute the ratio of the
area of the region to the area of the region obtained by scaling it by
a factor of c, where c > 0. For example, if c = 1/2, then to scale S by
a factor of c means that its edges now have length L/2. Sketch an
example in each case (square, circle, regular hexagon) of such a
region and scaled copy of each region. Your sketch should be
accurate (measure the distances) and you should state which value of
c you chose for your sketches.
Suppose that R is a planar figure enclosing a finite area. What do
you conjecture is the ratio of the area of R to the area of the figure
obtained by scaling R by a factor of c, where c > 0?

Explanation / Answer

scaling down by c in length reduces area by c2

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