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In the figure, the circle circumscribed about square ABCD has a circumference of \(16\pi\). The length of a side of square ABCD is s. What is the square’s area?
(A) \(8 \sqrt{2}\)
(B) 128
(C) 256
(D) \(128\pi\)
(E) \(256\pi\)
Attachment:
2018-04-24_1708.png [ 13.65 KiB | Viewed 2827 times ]
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Ans: B (128)
Perimeter is given of Circle by this we can get the diameter of the Circle which is diagonal of the square. and by this we can know the area of the square..
Perimeter is given of Circle by this we can get the diameter of the Circle which is diagonal of the square. and by this we can know the area of the square..
Bunuel
\(C=\pi*d=16*\pi\)
So, \(d=16\)
\(s\sqrt{2}=16\)
\(s=8\sqrt{2}\)
\(S=s^2=(8\sqrt{2})^2=128\)
Answer: B
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POE method:
Circle has a circumference of \(16\pi\) = \(2*pi*R\) => \(R=8\). Area of circle = \(3.14*8^2\)= 192
Now let's eliminate options 192<C<D<E, since inscribed square has to have area less than the circle's one. And we have to elemenate A, because \(8 \sqrt{2}\) = 11, which is the way too small (6% of the area of our circle).
(A) \(8 \sqrt{2}\)
(B) 128
(C) 256
(D) \(128\pi\)
(E) \(256\pi\)
Answer (B)
Circle has a circumference of \(16\pi\) = \(2*pi*R\) => \(R=8\). Area of circle = \(3.14*8^2\)= 192
Now let's eliminate options 192<C<D<E, since inscribed square has to have area less than the circle's one. And we have to elemenate A, because \(8 \sqrt{2}\) = 11, which is the way too small (6% of the area of our circle).
(A) \(8 \sqrt{2}\)
(B) 128
(C) 256
(D) \(128\pi\)
(E) \(256\pi\)
Answer (B)
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Bunuel
Since the circumference is 16π and C = πd , we see that the circle’s diameter = 16, which is also the length of the diagonal of the square.
Since diagonal = s√2, we have:
s√2 = 16
s = 16/√2
Thus, the area of the square is s^2 = (16/√2)^2 = 256/2 = 128.
Answer: B
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The length of the circumference (perimeter) is 16π.
The diameter of the circle is the diagonal of the circumscribed square.
Let us remember that:
16π = Diameter x π
So the diameter is 16, and the diagonal of the circumscribed square is also 16.
Let us remember that:
The area of a square of side S is Sexp2
In a square of side "A", the diagonal raised to 2 of the square is equal to 2 times the side squared.
In a square of side "S", 2 x Sexp2 = 16exp2
Then:
The area of the square with side S is:
Sexp2 = (16 x16)/2
= 8x16
A positive integer, whose unit is 8: ....8
Answer B
The diameter of the circle is the diagonal of the circumscribed square.
Let us remember that:
16π = Diameter x π
So the diameter is 16, and the diagonal of the circumscribed square is also 16.
Let us remember that:
The area of a square of side S is Sexp2
In a square of side "A", the diagonal raised to 2 of the square is equal to 2 times the side squared.
In a square of side "S", 2 x Sexp2 = 16exp2
Then:
The area of the square with side S is:
Sexp2 = (16 x16)/2
= 8x16
A positive integer, whose unit is 8: ....8
Answer B
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