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# In the figure above, the centers of four equal circles lie along the

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Math Expert
Joined: 02 Sep 2009
Posts: 59712
In the figure above, the centers of four equal circles lie along the  [#permalink]

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02 Feb 2016, 01:57
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Difficulty:

35% (medium)

Question Stats:

73% (01:33) correct 27% (01:07) wrong based on 126 sessions

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In the figure above, the centers of four equal circles lie along the diameter of the large circle. If the circumference of the large circle is 64π, what is the area of the shaded region?

A. 16π
B. 32π
C. 64π
D. 128π
E. 256π

Attachment:

2016-01-31_1804.png [ 4.24 KiB | Viewed 1406 times ]

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In the figure above, the centers of four equal circles lie along the  [#permalink]

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Updated on: 03 Feb 2016, 18:21
Editing my solution:

2*pi*r = 64*pi
r = 32

d1 + d2 + d3 + d4 = 2*32
Since d1 = d2 = d3 = d4; 4d1 = 64
d1 = 16
d1 = 2r1
r1 = 8

Area of one shaded circle = pi*(8^2) = 64pi
Area of four shaded circles = 4*64pi = 256pi

Originally posted by Vyshak on 02 Feb 2016, 20:54.
Last edited by Vyshak on 03 Feb 2016, 18:21, edited 1 time in total.
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Re: In the figure above, the centers of four equal circles lie along the  [#permalink]

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03 Feb 2016, 09:43
C = pi * D

D of large circle = 64
D of each black circle = 64/4 = 16
r of each black circle = 8

A = pi * r^2
A of each black circle = pi * 8^2 = 64 * pi

Total A of black circles = 4 * 64 * pi = 256 * pi

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Re: In the figure above, the centers of four equal circles lie along the  [#permalink]

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03 Feb 2016, 10:18
You are right. I made a foolish mistake by considering the radius as the diameter in the first step.
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Re: In the figure above, the centers of four equal circles lie along the  [#permalink]

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03 Feb 2016, 10:21
Bunuel wrote:

In the figure above, the centers of four equal circles lie along the diameter of the large circle. If the circumference of the large circle is 64π, what is the area of the shaded region?

A. 16π
B. 32π
C. 64π
D. 128π
E. 256π

Attachment:
2016-01-31_1804.png

Circumference of Larger Circle = πD = 64π
i.e. Diameter, D = 64
i.e. iameter of smaller circle, d = D/4 = 64/4=16
Radius of smaller circle = 16/2 = 8

Area of 4 smaller circles = 4*π*r^2 = 4*π*8^2 = 256π

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Re: In the figure above, the centers of four equal circles lie along the  [#permalink]

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03 Feb 2016, 10:22
Vyshak wrote:
2*pi*r = 64*pi
r = 32

d1 + d2 + d3 + d4 = 32
Since d1 = d2 = d3 = d4; 4d1 = 32
d1 = 8
d1 = 2r1
r1 = 4

Area of one shaded circle = pi*(4^2) = 16pi
Area of four shaded circles = 4*16pi = 64pi

The highlighted step above is incorrect

The correct equation should be d1 + d2 + d3 + d4 = Diameter of bigger circle = 64
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Re: In the figure above, the centers of four equal circles lie along the  [#permalink]

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07 Aug 2017, 15:21
R must be 32
since we have 4 smaller circles, R of each of them must be equal to 32/4 = 8.
area is pi*r^2.
64pi is area of a small circle
area of 4 small circles =64*4 or 128*2 or 256pi.

E.
Re: In the figure above, the centers of four equal circles lie along the   [#permalink] 07 Aug 2017, 15:21
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