Question

Question #6547a

Answer 1

`sqrt(2) : 1`

Explanation:

Let the radius of the inscribed (smaller circle) be `r` so it's circumference: `C_S = 2pi r`

This means the square has a length of `2r`

The radius of the circumscribed circle (larger circle) is half of the length of the diagonal of the square: `r_L = 1/2 d`

Using Pythagorean Theorem `c = sqrt(a^2 + b^2)` to find the length of the square's diagonal:

`d = sqrt((2r)^2 + (2r)^2) = sqrt (4r^2 + 4r^2) = sqrt (8r^2) = sqrt(8)sqrt(r^2) = sqrt(4 * 2) * r = 2r sqrt(2)`

`r_L = 1/2 d = (2r sqrt(2))/2 = r sqrt(2)`

`C_L = 2 pi r sqrt(2)`

Find the ratio of `C_L/C_S = (2 pi r sqrt(2))/ (2 pi r) = sqrt(2)/1 " or " sqrt(2) : 1`