Difference between revisions of "1952 AHSME Problems/Problem 33"

(Created page with "== Problem == A circle and a square have the same perimeter. Then: <math>\text{(A) their areas are equal}\qquad</math> <math>\text{(B) the area of the circle is the greater} ...")
 
 
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A circle and a square have the same perimeter. Then:  
 
A circle and a square have the same perimeter. Then:  
  
<math>\text{(A) their areas are equal}\qquad</math>
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<math>\text{(A) their areas are equal}\qquad\
<math>\text{(B) the area of the circle is the greater} \qquad</math>
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\text{(B) the area of the circle is the greater} \qquad\
<math>\text{(C) the area of the square is the greater} \qquad</math>
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\text{(C) the area of the square is the greater} \qquad\
<math>\text{(D) the area of the circle is }\pi\text{ times the area of the square}\qquad</math>
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\text{(D) the area of the circle is } \pi \text{ times the area of the square}\qquad\
<math>\text{(E) none of these}</math>
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\text{(E) none of these}</math>
  
 
== Solution ==
 
== Solution ==
<math>\fbox{}</math>
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 +
Assume that the side length of the square is 4 units. Then the perimeter for both shapes is is 4 * 4 = 16 units. Since the perimeter of a circle is 2πr, then 2πr =16 and r = 8/π, which is about 2.5. The area of a circle is πr^2 so (2.5^2)*π = 6.25π or about 19.6. This is more than the area of the square, which is 4^2 =16, so the answer is <math>\fbox{B}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 12:15, 14 September 2020

Problem

A circle and a square have the same perimeter. Then:

$\text{(A) their areas are equal}\qquad\\ \text{(B) the area of the circle is the greater} \qquad\\ \text{(C) the area of the square is the greater} \qquad\\ \text{(D) the area of the circle is } \pi \text{ times the area of the square}\qquad\\ \text{(E) none of these}$

Solution

Assume that the side length of the square is 4 units. Then the perimeter for both shapes is is 4 * 4 = 16 units. Since the perimeter of a circle is 2πr, then 2πr =16 and r = 8/π, which is about 2.5. The area of a circle is πr^2 so (2.5^2)*π = 6.25π or about 19.6. This is more than the area of the square, which is 4^2 =16, so the answer is $\fbox{B}$.

See also

1952 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 32
Followed by
Problem 34
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