Question #6547a

1 Answer
Apr 5, 2017

#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#