Question 237228: If the perimeter of a circle is equal to that of a square, then the ratio of their
areas is
(A) 22 : 7 (B) 14 : 11 (C) 7 : 22 (D) 11: 14
Found 2 solutions by rapaljer, Theo:
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! I answered this question a few minutes ago, but I did not have the multiple choices that you gave in this problem. To see my solution, look for Algebra.com problem #237229. In that solution the ratio of the area of the square to the area of the circle came out to  . Apparently, those who posed this question used the value of  , so my answer
=
=
= , which is your answer D.
HOWEVER, in this question, you were NOT clear as to whether you wanted the ratio of the area of the SQUARE to the CIRCLE, or the CIRCLE to the SQUARE. In the previous problem that I solved, it was clearly stated the ratio of the area of the SQUARE to the area of the CIRCLE, so this would be the answer -- D.
If, in fact, you wanted the ratio of the SQUARE to the CIRCLE, the answer is reversed which would be 14 to 11, which is answer B. Which choice did you want?? By the way, I spent a LOT of time on this question, but it's a neat problem and well worth the effort!!
Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the ratio appears to be 11/14 or 14/11 depending on who's on top.
if it's the ratio of the area of the circle to the area of the square, then the ratio is 14 to 11 = 14/11
since they mentioned the circle first, i would assume they want the ratio of the area of the circle to the area of the square.
that would be 14 : 11 which is equivalent to 1.2727272727.....
here's how.
statement is that the perimeter of the circle equals the perimeter of the square.
that means that:
4*s = 2*pi*r
divide both sides by 2*pi to get:
r = 4*s/2*pi = 2*s/pi
area of a circle is equal to pi*r^2
area of a square is equal to s^2
ratio of the area of a circle to a square is pi*r^2/s^2
since r = 2*s/pi that becomes:
ratio of the area of a circle to a square is pi*(2*s/pi)^2 / s^2
that becomes:
(pi*(2*s)^2)/(pi)^2 / s^2 which becomes:
(4*s^2/pi) / s^2 which becomes:
4/pi
the ratio of the area of the circle to the area of the square is 4/pi which is equivalent to 1.273239545
1.273239545 is very close to 1.2727272727....
the differnce is in the rounding.
if you take 4/pi * 11 you will get 14.00563499
14/11 was really 14.00563499 but was truncated to the nearest digit.
none of the other answers even came close.
i would go with 14 / 11 as the correct ratio.
that would be answer (B)
11 / 14 could be an answer if the ratio was the area of the square to the area of the circle.
they did not specify directly.
they only implied, by mentioning the circle first.
the question should have read:
If the perimeter of a circle is equal to that of a square, then the ratio of the area of the circle to the square is:
(A) 22 : 7 (B) 14 : 11 (C) 7 : 22 (D) 11: 14
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