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The circumference of a circle is

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The circumference of a circle is [#permalink] New post 15 Sep 2017, 08:42
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The circumference of a circle is \(\frac{7}{8}\) the perimeter of a square.

Quantity A
Quantity B
The area of the square
The area of the circle


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Last edited by Carcass on 04 Oct 2017, 05:38, edited 2 times in total.
edit the OA
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Re: The circumference of a circle is [#permalink] New post 16 Sep 2017, 08:01
My solution is: since we now that the circumference of the circle is 7/8 the perimeter of the square, we can set 2*pi*r=7/8*4*l, from which we can derive that r = 7/(4*pi)*l. Then the area of a square is l^2, whereas the area of a circle is pi*r^2, where we can substitute r as above. Then, we have to compare l^2 to 49/(16*pi)*l^2 and since 49/16*pi is less than one, the area of the square is greater. So answer should be A. Why B?
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Re: The circumference of a circle is [#permalink] New post 16 Sep 2017, 15:45
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IlCreatore wrote:
My solution is: since we now that the circumference of the circle is 7/8 the perimeter of the square, we can set 2*pi*r=7/8*4*l, from which we can derive that r = 7/(4*pi)*l. Then the area of a square is l^2, whereas the area of a circle is pi*r^2, where we can substitute r as above. Then, we have to compare l^2 to 49/(16*pi)*l^2 and since 49/16*pi is less than one, the area of the square is greater. So answer should be A. Why B?


OE

Quote:
Because the circumference of a circle depends on π(C = πd), it is best to pick values for the square. If the side of the square is 2, the perimeter is 4(2) = 8 and the area is (2)(2) = 4. Then, circumference of the circle is \(\frac{7}{8} * 8 =7\) Since circumference is 2πr = 7, the radius of the circle is r = 7/2π

Quantity A: The area of the square = 4.

Quantity B: The area of the circle = πr^2 = π (7/2π)^2 = π (7/2π)^2 = π (49/4π^2)= 49/4 π= about 3.9





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Re: The circumference of a circle is [#permalink] New post 21 Sep 2017, 08:02
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Carcass wrote:
IlCreatore wrote:
My solution is: since we now that the circumference of the circle is 7/8 the perimeter of the square, we can set 2*pi*r=7/8*4*l, from which we can derive that r = 7/(4*pi)*l. Then the area of a square is l^2, whereas the area of a circle is pi*r^2, where we can substitute r as above. Then, we have to compare l^2 to 49/(16*pi)*l^2 and since 49/16*pi is less than one, the area of the square is greater. So answer should be A. Why B?


OE

Quote:
Because an exterior angle of a triangle is equal to the sum of the two opposite interior angles of the triangle (in this case, the top small triangle), c = a + b.
Therefore, d > c and a + b = c taken together imply that d > a + b. Subtract b from both sides: d – b > a. Quantity B is greater.


Triangles? Should I divide the square in two triangles? I really don't get how a triangle matters in a comparison between a circle and a square
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Re: The circumference of a circle is [#permalink] New post 21 Sep 2017, 20:05
Expert's post
IlCreatore wrote:
Carcass wrote:
IlCreatore wrote:
My solution is: since we now that the circumference of the circle is 7/8 the perimeter of the square, we can set 2*pi*r=7/8*4*l, from which we can derive that r = 7/(4*pi)*l. Then the area of a square is l^2, whereas the area of a circle is pi*r^2, where we can substitute r as above. Then, we have to compare l^2 to 49/(16*pi)*l^2 and since 49/16*pi is less than one, the area of the square is greater. So the answer should be A. Why B?


OE

Quote:
Because an exterior angle of a triangle is equal to the sum of the two opposite interior angles of the triangle (in this case, the top small triangle), c = a + b.
Therefore, d > c and a + b = c taken together imply that d > a + b. Subtract b from both sides: d – b > a. Quantity B is greater.


Triangles? Should I divide the square into two triangles? I really don't get how a triangle matters in a comparison between a circle and a square


Sorry for the mismatch. My pdf is so tight that I wrote the explanation of the previous question. Apologize.

Fixed both the OA and OE
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Re: The circumference of a circle is   [#permalink] 21 Sep 2017, 20:05
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