Question 885348
volume of cylinder is equal to pi * r^2 * h
volume of rectangular prism is equal to l * w * h
l is the length
w is the width
h is the height
since w = 2 * r, the volume of the rectangular prism becomes equal to l * 2 * r * h.
let v1 = volume of the cylinder
let v2 = volume of the prism.
you get:
v1 = pi * r^2 * h
v2 = l * 2 * r * h
ratio of the cylinder to the prism is equal to v1/v2 which becomes:
v1/v2 = (pi * r^2 * h) / (l * 2 * r * h)
simplify this to get:
v1/v2 = (pi * r) / (2 * l)
r is the radius of the cylinder, l is the length of the prism.
that's your ratio.


you can confirm by making a test with some numbers.
assume r = 3 and h = 9
volume of the cylinder is equal to pi * r^2 * h = pi * 9 * 9 = pi * 81.
assume l = 5, w = 6, h = 9 for the prism.
the width of 6 is equal to 2 times the radius of 3.
volume of the prism is equal to 5 * 6 * 9 = 270
ratio of the volume of the cylinder to the prism is equal to (pi * 81) / 270.
divide numerator and denominator of this fraction by 27 and you get (pi * 3) / 10.
since 10 is equal to 2 * 5, the ratio becomes:
(pi * 3) / (2 * 5)
since 3 is the radius and 5 is the length, then this ratio is equivalent to:
(pi * r) / (2 * l) which confirms the formula for the ratio is correct.