The ratio to the perimeter of the circle to the square of the circle is 2:3 . Find the ratio to the?

Shouldn't this read:

The ratio of the perimeter of a circle to the perimeter of a square is 2:3 .
Find the ratio of the area of the circle to the area of the square.

P.c = perimeter of circle
P.s = perimeter of square
r = radius of circle
b = side of square
A.c = area of circle
A.s = area of square

P.c = 2πr = 2
r = 2 / (2π)
r = 1/π.

A.c = πr²
A.c = π (1/π)²
A.c = π * 1/π²
A.c = 1/π.

P.s = 4b = 3
b = 3/4.

A.s = b² = (3/4)² = 9/16.

The ratio of A.c : A.s is then:

A.c / A.s = (1/π) / (9/16)
A.c / A.s = (1/π) * (16/9)

A.c / A.s = 16 / 9π.
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So, it is not the case that
"The ratio of areas is always the square of the ratio of sizes, so the answer is 4:9."

But The ratio of areas is always PROPORTIONAL TO the square of the ratio of sizes, so the PROPORTIONALITY FACTOR is 4:9.

A square is not a circle, the same perimeters of different shapes enclose different areas. In our example, let's take the same perimeter

P.c = P.s = 1

r = 1 / (2π)
A.c = 1 / (4π)

b = 1/4
A.s = 1/16

The ratio of the shapes [circle : square] is then
A.c / A.s = 4/π.

Applying the proportionality factor yields our original result:

(4/π) * (4/9) = 16 / 9π.