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Intern  Joined: 17 May 2013
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13 00:00

Difficulty:   35% (medium)

Question Stats: 72% (02:05) correct 28% (01:56) wrong based on 437 sessions

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Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?
Attachment:
File comment: This is the figure for the question. GeometryPost12Ques2.jpg [ 7.89 KiB | Viewed 54867 times ]

(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

Originally posted by genuinebot85 on 24 Jul 2013, 11:14.
Last edited by Bunuel on 24 Jul 2013, 11:17, edited 1 time in total.
Edited the question.
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Joined: 02 Sep 2009
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Re: Four identical circles are drawn in a square such that each  [#permalink]

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genuinebot85 wrote:
Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region? (A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

Look at the image below:
Attachment: Untitled.png [ 13.57 KiB | Viewed 41427 times ]
The areas of regions with red dots are equal. So, we have 16 equal regions and we need the area of four of them. The area of all 16 is equal to the area of the square minus the area of four circles.

The area of the square = $$20^2 = 400$$.
The area of four circles = $$4*(\pi{r^2})=4*(\pi{5^2})=100\pi$$ (the diameter of each circle is 1/2 of the side, thus the radius of each circle is 1/4 of the side).

The area of 16 regions = $$400-100\pi$$.
The area of shaded region (4 regions with red dots) = $$\frac{400-100\pi}{4}=100-25\pi$$.

Else you could simply find the area of the smaller square (1/4 of the bigger) and subtract the area of the circle. This way you'd also get the area of 4 regions with red dots.

Hope it's clear.
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Re: Four identical circles are drawn in a square such that each  [#permalink]

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genuinebot85 wrote:
Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?
Attachment:
GeometryPost12Ques2.jpg

(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

You can draw a second square, with vertices at the centers of the circles. Then that square has sides of 10 units, and the quarter circles have total area of 25*pi.
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The shaded region in the problem is equal to the shaded region in the modified diagram (in red) as shown in diagram below

Area of square$$= 10^2 = 100$$

Area of Circle $$= \pi (\frac{10}{2})^2 = 25\pi$$

Area of red shaded region$$= 100 - 25\pi$$

Attachments GeometryPost12Ques2.jpg [ 9.25 KiB | Viewed 40380 times ]

Intern  Joined: 17 Apr 2015
Posts: 5
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Hi everyone, I got this one right on an educated guess, having successfully whittled down the options to C or D, but I'm still a little hazy on why we ultimately end up dividing by four instead of two (ie 400 - 100pi divided by 4 = correct answer C as opposed to 400 - 100pi divided by 2 which would yield answer choice B). New to this so I hope Im not confusing anyone! Thanks in advance!
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MeliMeds wrote:
Hi everyone, I got this one right on an educated guess, having successfully whittled down the options to C or D, but I'm still a little hazy on why we ultimately end up dividing by four instead of two (ie 400 - 100pi divided by 4 = correct answer C as opposed to 400 - 100pi divided by 2 which would yield answer choice B). New to this so I hope Im not confusing anyone! Thanks in advance!

$$400-100\pi$$ is the area of 16 regions.

We need the area of 4 shaded region, thus we need to divide $$400-100\pi$$ by 4: $$\frac{400-100\pi}{4}=100-25\pi$$.

Hope it's clear
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Bunuel wrote:
MeliMeds wrote:
Hi everyone, I got this one right on an educated guess, having successfully whittled down the options to C or D, but I'm still a little hazy on why we ultimately end up dividing by four instead of two (ie 400 - 100pi divided by 4 = correct answer C as opposed to 400 - 100pi divided by 2 which would yield answer choice B). New to this so I hope Im not confusing anyone! Thanks in advance!

$$400-100\pi$$ is the area of 16 regions.

We need the area of 4 shaded region, thus we need to divide $$400-100\pi$$ by 4: $$\frac{400-100\pi}{4}=100-25\pi$$.

Hope it's clear

Thanks! I actually pondered over it some more and found that my brain more easily accepted the version of splitting the figure into four even squares... ie the area of one square minus the area of its internal circle divided by 4 would give the area of one of the four middle segments, then multiplying that by four equates the total central portion (which is the same as multiplying area of one square minus its internal circle by one since the fours cancel out).. I know both strategies get you to the same answer but I guess that's the beauty and diversity of perspective in learning....some people just see things in certain ways that others see in others toward the same net result....THANKS ALL THE SAME...it helps a lot to reason things out!
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Re: Four identical circles are drawn in a square such that each  [#permalink]

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Hi All,

Another way to solve the question, imagine a square joining the centres of 4 circles. See picture below.

You know the radius=5. So the area of new square= 100.

You have to substract the 4 sectors. The area of sector= PI R*R*angle/360= PI* 5 *5 /4 = 25 PI/4 (multiplying by four, because we have 4 sectors)

=25PI. ( Angle subtended at the centre=90 and hence we took the angle 90)

So area of shaded region = 100-25PI.

genuinebot85 wrote:
Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?
Attachment:
The attachment GeometryPost12Ques2.jpg is no longer available

(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

Attachments GeometryPost12Ques2.jpg [ 8.75 KiB | Viewed 39572 times ]

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Re: Four identical circles are drawn in a square such that each  [#permalink]

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genuinebot85 wrote:
Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?
Attachment:
GeometryPost12Ques2.jpg

(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

we can see that regions not shaded can be "glued" to form 4 identical forms to the center one (shaded region)
area of square = 20^2 = 400
area of 1 circle = 25pi => 4 circles = 100pi.
now..
400-100pi = area of everything except the circles.
since we can imaginably draw 4 identical to central one figures, we divide everything by 4:
100-25pi.

C
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Four identical circles are drawn in a square such that each  [#permalink]

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Bunuel wrote:
genuinebot85 wrote:
Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region? (A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Could not understand the solution, need help.

Look at the image below:
Attachment:
Untitled.png
The areas of regions with red dots are equal. So, we have 16 equal regions and we need the area of four of them. The area of all 16 is equal to the area of the square minus the area of four circles.

The area of the square = $$20^2 = 400$$.
The area of four circles = $$4*(\pi{r^2})=4*(\pi{5^2})=100\pi$$ [b](the diameter of each circle is 1/2 of the side, thus the radius of each circle is 1/4 of the side)[/b].

The area of 16 regions = $$400-100\pi$$.
The area of shaded region (4 regions with red dots) = $$\frac{400-100\pi}{4}=100-25\pi$$.

Else you could simply find the area of the smaller square (1/4 of the bigger) and subtract the area of the circle. This way you'd also get the area of 4 regions with red dots.

Hope it's clear.

Hi Bunuel,

Do you mind elaborating more on the red part ? Thanks !
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Re: Four identical circles are drawn in a square such that each  [#permalink]

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Area of a square - 400 sq cm

Draw another square from center of each circle
Area of small square - 100 sq cm

Radius of circle - 5 cm (20 cm is the side length, half of it is diameter and further half of 20 is radius)
Area of 4 sectors - 4*90/360 pi 5*5 = 25pi

Area of shaded region = area of small square - area of 4 minor sectors
= 100-25pi Ans C
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Re: Four identical circles are drawn in a square such that each  [#permalink]

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I found it easier to realise that there are actually 4 intershape sub-sections (See diagram).

We can find the area of the square (400), deduct the 4x equal circle areas (5^2*pi)*4 =100 pi then divide by 4 to get just one intershape sub-section.
Attachments Capture.JPG [ 27.84 KiB | Viewed 1737 times ] Re: Four identical circles are drawn in a square such that each   [#permalink] 11 Sep 2019, 15:33
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