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If the four circles shown above touch two other circles at exactly one

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If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 13 Sep 2018, 04:47
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If the four circles shown above touch two other circles at exactly one point each and have a radius of 5, and the centers of each pair of opposite circles are the same distance apart, what is the approximate area of the shaded region?

A. 76
B. 50
C. 36
D. 25
E. 22

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If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post Updated on: 15 Sep 2018, 09:02
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Area of the shaded region = area of the square with vertex at the center - area of 4 quarters of the circle (or one circle)

Area of the circle\(=πr^2=25π\)

Side of the square with Vertex at the center of the circle side of the square = 2r = d

Area of the square with Vertex \(= d^2*d^2 = 100\)

\(100-(25*22)/7=(700-550)/7 = 150/7 = 21.4 = approx 22\)

Originally posted by AvidDreamer09 on 15 Sep 2018, 01:23.
Last edited by AvidDreamer09 on 15 Sep 2018, 09:02, edited 3 times in total.
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If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 15 Sep 2018, 06:58
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Attachment:
SquareCircle.jpg
SquareCircle.jpg [ 10.75 KiB | Viewed 931 times ]


Area(Shaded region) = Area(Square with side 10) - Area(4 quarter circles or 1 circle with radius 5)

Formula used:
Area of a square = \(side^2\)
Area of a quarter circle = \(\frac{1}{4} * \pi * r^2\) where r - radius

Substituting values, Area(Square) = \(10^2 = 100\)
Area of each quarter circle = \(\frac{1}{4} * \pi * 5^2 = \frac{1}{4} * \frac{22}{7} * 25 = \frac{275}{14} = 19.5\)(apprx)
Since we have four quarter circles, the total area enclosed by them is \(4*19.5 = 78\)

Therefore, the area of the shaded region which is the difference in areas is 100 - 78 = 22(Option E)
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Re: If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 15 Sep 2018, 11:51
If the four circles shown above touch two other circles at exactly one point each and have a radius of 5, and the centers of each pair of opposite circles are the same distance apart, what is the approximate area of the shaded region?

A. 76
B. 50
C. 36
D. 25
E. 22

Lets first take the area of the square formed by joining the centers of the circles, the side of teh square will be 10 units each

total area=10*10=100.

now we see that there are 4 sectors in the square ,all angles of the square are 90degree, so the total is 4*90=360.(equal to an entire circle.)

so we find the area of the circle, which is PI*5*5=25PI=78.something.

100-area of the circle we get aprox 22.

E is the correct option.
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If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post Updated on: 16 Sep 2018, 01:53
pushpitkc wrote:
Attachment:
SquareCircle.jpg


Area(Shaded region) = Area(Square with side 10) - Area(4 quarter circles or 1 circle with radius 5)

Formula used:
Area of a square = \(side^2\)
Area of a quarter circle = \(\frac{1}{4} * \pi * r^2\) where r - radius

Substituting values, Area(Square) = \(10^2 = 100\)
Area of each quarter circle = \(\frac{1}{4} * \pi * 5^2 = \frac{1}{4} * \frac{22}{7} * 25 = \frac{275}{14} = 19.5\)(apprx)
Since we have four quarter circles, the total area enclosed by them is \(4*19.5 = 78\)

Therefore, the area of the shaded region which is the difference in areas is 100 - 78 = 22(Option E)


hey pushpitkc :) i am back :lol: how did you get \(\frac{22}{7}\) ? :? what it means ?

thanks :-)

got it :)\(\frac{3.14*25}{4} = 19.5\)

Originally posted by dave13 on 15 Sep 2018, 12:20.
Last edited by dave13 on 16 Sep 2018, 01:53, edited 1 time in total.
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Re: If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 15 Sep 2018, 18:48
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10 seconds approach:

The formula for finding the area of the shaded region for this specific figure is r^2(4-pi). In this case, it would be 25(4-pi) which roughly comes out to be 21. something. Click option E and you are done!
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If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 16 Sep 2018, 01:58
csaluja wrote:
10 seconds approach:

The formula for finding the area of the shaded region for this specific figure is r^2(4-pi). In this case, it would be 25(4-pi) which roughly comes out to be 21. something. Click option E and you are done!



hey csaluja, nice approach :)

\(r^2(4-pi)\) can i apply this formula to any kind of figures ?

how about formulas of shaded regions of triangles, squares, quadrilaterals etc ? any similar 10sec approach formulas ?

thanks!
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Re: If the four circles shown above touch two other circles at exactly one  [#permalink]

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New post 16 Sep 2018, 15:20
dave13 wrote:
csaluja wrote:
10 seconds approach:

The formula for finding the area of the shaded region for this specific figure is r^2(4-pi). In this case, it would be 25(4-pi) which roughly comes out to be 21. something. Click option E and you are done!



hey csaluja, nice approach :)

\(r^2(4-pi)\) can i apply this formula to any kind of figures ?

how about formulas of shaded regions of triangles, squares, quadrilaterals etc ? any similar 10sec approach formulas ?

thanks!



Hello Dave,

No, this formula is strictly for the given figure's shaded area. It will not work with other figures's shaded area. Yes, there are multiple formulas like this. I would recommend reading the Geometry chapter from Nishit Sinha's Quant Aptitude CAT Book. I learnt this formula from that chapter!

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Re: If the four circles shown above touch two other circles at exactly one   [#permalink] 16 Sep 2018, 15:20
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