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

The area of a circle with a diameter d is
`( \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 : ( \pi d^2)/(4)`
Since there is a common term on both sides, we cancel
`d^2` and get the final ratio of:

`1: ( \pi )/(4)`

Answer 2

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