Question

A circle is inscribed in a square. Write and simplify an expression for the ratio of the area of the square to the area of the circle. For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square.

Answer 1

Given:
circle inscribed in a square.
Side length of the square = diameter of the circle.
Let x side length and diameter.

Area of a square = x²

Area of a circle = πr²

r = radius ; half of the diameter. = x/2

Area of a circle = π * (x/2)²  or π (x²/4)

Ratio of the area of the square to the area of the circle

x² : π(x²/4)   or x² / πx²/4

x² * 4/πx² = 4/π

Answer 2

The area of a circle with a diameter d is $\frac{ \pi d^4}{4}$ and the area of a square whose side is the diameter d of the circle is $d^2$.
The ratio of the area of a square to the area of a circle is
$d^2 : \frac{ \pi d^2}{4}$
Since there is a common term on both sides, we cancel $d^2$ and get the final ratio of:
$1: \frac{ \pi }{4}$