Question #6547a

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Answer 1 · 2017-04-05T16:11:59

$sqrt(2) : 1$

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 $C_L/C_S = (2 pi r sqrt(2))/ (2 pi r) = sqrt(2)/1 " or " sqrt(2) : 1$