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# For each of the three circles shown above, the diameter of the smaller

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Math Expert
Joined: 02 Sep 2009
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For each of the three circles shown above, the diameter of the smaller  [#permalink]

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26 Dec 2014, 08:43
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5% (low)

Question Stats:

86% (01:33) correct 14% (01:36) wrong based on 174 sessions

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Tough and Tricky questions: Geometry.

For each of the three circles shown above, the diameter of the smaller circle is equal to the radius of the larger. If the radius of the largest circle is equal to 8 inches, what is the area of the shaded region in square inches?

A. 3π
B. 6π
C. 12π
D. 15π
E. 18π

Kudos for a correct solution.

Attachment:

2014-12-26_1941.png [ 3.5 KiB | Viewed 5279 times ]

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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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26 Dec 2014, 09:27
Quote:
For each of the three circles shown above, the diameter of the smaller circle is equal to the radius of the larger. If the radius of the largest circle is equal to 8 inches, what is the area of the shaded region in square inches?

A. 3π
B. 6π
C. 12π
D. 15π
E. 18π

Starting from the largest circle, we first calculate the radii of the smaller circles:

We are told that the largest radius is 8 inches: $$r_{3}=8$$. From this follows $$r_{2}=4$$ and $$r_13}=2$$. ($$r_{1}$$ is the radius of the smallest circle and $$r_{2}$$ the radius of the middle circle.

The shaded area is then $$\pi * r_{2}^2 - \pi * r_{1}^2 = \pi * (r_{2}^2 - r_{1}^2) = \pi * 12 = 12\pi$$.

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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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26 Dec 2014, 17:49
Bunuel wrote:

Tough and Tricky questions: Geometry.

For each of the three circles shown above, the diameter of the smaller circle is equal to the radius of the larger. If the radius of the largest circle is equal to 8 inches, what is the area of the shaded region in square inches?

A. 3π
B. 6π
C. 12π
D. 15π
E. 18π

Kudos for a correct solution.

Attachment:
2014-12-26_1941.png

Say circles as A B C with Smallest is A and Largest is C

Radius of C = 8; B=4;C=2

Area of Shaded region = Area of B-Area of A
= 16 π - 4 π
= 12 π

Ans C
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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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27 Dec 2014, 00:41
if radius of largest circle is 8 then the medium circle's radius will be 4 and smallest circle radius will be 2.

Now area of circle = Raidus(r)*(r)*pie

hene area of shaded region= 4*4*pie- 2*2*pie= 12pie. hence answer is C
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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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31 Dec 2014, 21:50

Radius of large circle = 8, then

radius of medium circle = 4, and

radius of small circle = 2

Area of shaded region = Area of Medium circle - Area of small circle

$$= 16\pi - 4\pi = 12\pi$$
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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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08 Jan 2015, 09:15
Bunuel wrote:

Tough and Tricky questions: Geometry.

For each of the three circles shown above, the diameter of the smaller circle is equal to the radius of the larger. If the radius of the largest circle is equal to 8 inches, what is the area of the shaded region in square inches?

A. 3π
B. 6π
C. 12π
D. 15π
E. 18π

Kudos for a correct solution.

Attachment:
2014-12-26_1941.png

OFFICIAL SOLUTION:

(C) The first step is to understand the strategy. We have to find the area of the entire black circle and subtract from it the area of the smallest circle.

Since we know that the largest circle has a radius of 8, the black circle must have a radius of 4 and the smallest circle must have a radius of 2. The area of the smallest circle is equal to:
πr² = π2² = 4π.

Now, we want to find the area of the black circle:
πr² = π4² = 16π.

Finally, we subtract the smaller area from the area of the black circle to get 16π – 4π = 12π, or answer choice (C).
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Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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30 Jul 2017, 20:53
Can someone please correct me because am not understanding as to where am i going wrong in my steps. The area of the shaded part= the total area of the largest circle - area of the smallest circle. Am i correct. if yes then , The radius of the largest circle is 8 and hence the area is 64 pi. Where am i going wrong?kindly help.
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Posts: 54496
Re: For each of the three circles shown above, the diameter of the smaller  [#permalink]

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30 Jul 2017, 21:02
1
longhaul123 wrote:
Can someone please correct me because am not understanding as to where am i going wrong in my steps. The area of the shaded part= the total area of the largest circle - area of the smallest circle. Am i correct. if yes then , The radius of the largest circle is 8 and hence the area is 64 pi. Where am i going wrong?kindly help.

That's not correct. When you subtract the area of the smallest circle from the area of the largest circle you do not get the shaded region. Check the image below, you will get black region + red region, while we need only black region.
Attachment:

Untitled.png [ 3.15 KiB | Viewed 2357 times ]

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Re: For each of the three circles shown above, the diameter of the smaller   [#permalink] 30 Jul 2017, 21:02
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