Question

Mrs. Smith wants a circular patio with a diameter of 12 feet and Mr. Smith wants a square patio with a side of 12 feet. Which would have a larger area and by how much?

Answer 1

Mr. Smith's square is bigger than Ms. Smith's circle by `36(4-pi) "ft"^2`

Explanation:

The area of a circle of radius `r` is given by
`A_"circle" = pir^2`

The area of a square of side length `s` is given by
`A_"square" = s^2`

As the diameter of a circle is twice its radius, Ms. Smith wants a circle of radius `12/2 = 6`. Thus the area of Ms. Smith's circle is

`A_"circle" = pi6^2 = 36pi`

Mr. Smith wants a square of side length `12` and so the area of Mr. Smith's square is

`A_"square" = 12^2 = 144`

But `pi ~~ 3.14 < 4`
so
`A_"square" = 144 = 36 * 4 > 36*pi = A_"circle"`

Thus the square is bigger, and the difference between the two is

`A_"square" - A_"circle" = 144 - 36pi = 36(4-pi)`

Thus Mr. Smith's square is bigger than Ms. Smith's circle by `36(4-pi) "ft"^2`