Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58954

Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 07:59
Question Stats:
62% (01:57) correct 38% (02:26) wrong based on 205 sessions
HideShow timer Statistics
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region? A. \(12π\) B. \(122π\) C. \(12+π\) D. \(12+2π\) E. \(244π\)
Attachment:
Untitled.png [ 3.07 KiB  Viewed 2601 times ]
Official Answer and Stats are available only to registered users. Register/ Login.
_________________



Senior Manager
Joined: 10 Aug 2018
Posts: 338
Location: India
Concentration: Strategy, Operations
WE: Operations (Energy and Utilities)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 08:31
IMO A is the answer. area of the square is 4x4=16 shaded region is approx half, 8 12Pie is the only answer nearest to 8.
_________________
On the way to get into the Bschool and I will not leave it until I win. WHATEVER IT TAKES. " I CAN AND I WILL"
GMAT:[640 Q44, V34, IR4, AWA5]



Manager
Joined: 26 Jan 2016
Posts: 180

Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
Updated on: 23 Jul 2019, 08:51
Quote: Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
A. 12−π B. 12−2π C. 12+π D. 12+2π E. 24−4π Looking at the figure, we can say area of shaded region <50% of area of square but not very less than 50% Area of square=16 50% of 16 =8 So, any value less than 8 would be the area of the shaded region. Value of π~3.14 Now replacing in choices A. 12−π = 123.14=7.86 B. 12−2π =12(~)6= 6 C. 12+π >12. Eliminate D. 12+2π >12. Eliminate E. 24−4π = 24 (~)12= ~12. Eliminate B/w A &B, A is the closet. Hence A
_________________
Your Kudos can boost my morale..!!
I am on a journey. Gradually I'll there..!!
Originally posted by kitipriyanka on 23 Jul 2019, 08:36.
Last edited by kitipriyanka on 23 Jul 2019, 08:51, edited 1 time in total.



Manager
Joined: 10 Mar 2019
Posts: 75
Location: Russian Federation
GPA: 3.95

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 08:46
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
I divided the square into 4 small squares.
Square #1 (I quadrant)  its area is 2*2=4 and we need to count it. Squares #2 and #4 (II quadrant and III quadrant) if we put them together then we can see semicircle and we need to calculate eternal area 2*4  (Pi*2^2)/2=82Pi Square #3 we don't need to count this area is the answer.
Thus, 4+82Pi= 122Pi
The answer is B



Director
Joined: 28 Jul 2016
Posts: 637
Location: India
Concentration: Finance, Human Resources
GPA: 3.97
WE: Project Management (Investment Banking)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 08:49
Consider only one semicircle Its area will \(\frac{\pi}{2} *2^2\) area of the triangle will \(\frac{1}{2} *2\sqrt{2}*2\sqrt{2}\) thus area of those two leaves will be = \(\frac{\pi}{2} *2^2\)  \(\frac{1}{2} *2\sqrt{2}*2\sqrt{2}\) = \(2\pi  4\) Area of two semicircles without overall = \(4\pi\)  \(2\pi + 4\) = \(2\pi +4\)  eq 1 Total area of square = \(4^2 = 16\) subtract eq1 from the square area to get shaded part = \(16  2\pi +4\) = \(12  2\pi\) = B answer
Attachments
imagegmat.png [ 7.43 KiB  Viewed 2108 times ]



Intern
Joined: 10 May 2018
Posts: 17

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 08:50
I wasn't sure how to solve for the section between the two circles which I knew needed to be added back in, so I approximated. The area of the square would be 16. The shaded section is approximately only slightly less than half the area, which would be 8. π ~= 3
A. 12−π 12  3 = 9  This is more than half
B. 12−2π 12  2*3 = 6  This seems to be the closest answer
C. 12+π 12 + 3 = 16  This is basically the size of the square so cannot be the answer
D. 12+2π 12+2*3 = 18  This is larger than the area of the square so cannot be the answer
E. 24−4π 24  4*3 = 12  This is too big



Senior Manager
Joined: 05 Mar 2017
Posts: 261
Location: India
Concentration: General Management, Marketing
GPA: 3.6
WE: Marketing (Hospitality and Tourism)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 08:55
Image Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
To answer the question you need to deduct the area of the Two semicircles from the area of the Square. Area of the whole square is 4*4 = 16 The area of the semicircle is Pie/2 *R^2, R is equal to two. This is going to be multiplied by 2 as there are 2 semicircles, but then we are counting the part of the semicircle which is common to both twice, hence we need to deduct that.
16  4pie  4 2Pie = 12 2pie.
Hence the answer is option B.
A. 12−π B. 12−2π C. 12+π D. 12+2π E. 24−4π
B



Director
Joined: 22 Nov 2018
Posts: 562
Location: India
GMAT 1: 640 Q45 V35 GMAT 2: 660 Q48 V33

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:00
Area of the square  4*4=16 Area of the 2 semi circles = (2*π*2*2)/2 = 4π Area of the common segment between 2 semi circles = (Area of 4 semi circles  Area of the square)/4 <when 4 semicircles are made within the square 4 common segments similar to the one in the diagram are created and leave zero uncovered area> = ((4*π*2*2)/2  16)/4 = (8π  16)/4 = 2π4 Area of shaded region = Area of square  (area of 2 semicircle  area of the common segment) = 16(4π(2π4) = 164π+2π4 = 122π IMO B
_________________
Give +1 kudos if this answer helps..!!



Manager
Joined: 30 Aug 2018
Posts: 103
Location: India
Concentration: Finance, Accounting
GPA: 3.36
WE: Consulting (Computer Software)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:01
Area of square =16 area of shaded region is almost half around 8
option 1 is the closest.



Intern
Joined: 26 May 2018
Posts: 45

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:05
Answer is A.
The area of the semicircle is A = pi * r^2/2 since we have two semicircles then the total area is equal to the area of a cicle. A = pi * r^2 Circumference is 2*pi*r



Manager
Joined: 26 Mar 2019
Posts: 100
Concentration: Finance, Strategy

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:08
https://gmatclub.com/forum/download/file.php?id=49179Quote: Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
This question is a bit tricky. We have a square with side length of 4 and two semicircles in it. We need to find the area of space not occupied by semicircles. Let us first of all, divide our square into 4 equal pieces and numerate them 1 to 4 from upper and left one to lower and right one. Let us look at these small circles one by one. First of all, the area of such a small circle is \(S=(a/2)^2=(4/2)^2=4\) Also, we might notice that the diameter of a semicircle is equal to the side of a bigger square. Thus, \(r=d/2=a/2=2\) The area of semicircle is \(S=(pi*r^2)/2=(pi*2^2)/2=2*pi\) Now, let us have a closer look at these squares: 1) Upper left square. We can see that there is a half of semicircle in this small square. The area of the half of semicircle is \(S=2*pi/2=pi\) So the area of the shaded region will be \(area of small square  area of half of semicircle = 4  pi\) 2) Upper right square This small square does not have its area covered by a semicircle, thus its all area is shaded and is equal to 4 3) Lower left square This square is covered by 2 halves of semicircles and we can notice that it does not have any area shaded. Thus, its shaded area is 0. 4) Lower right square Here, as in the first example, the shaded area is equal to \(area of small square  area of half of semicircle = 4  pi\) Adding all these areas together we get: \((4  pi) + 4 + 0 + (4  pi) = 12  2*pi\) Answer: B



Senior Manager
Joined: 12 Dec 2015
Posts: 436

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:13
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
Solution: If we draw 4 semicycles on each one of the sides, then all the semicycles will meet/intersect in the center(crossing point of the square's diagonals). So (the area of the square) = 4 * (each semicycle area)  4 * (area of the each eyeshaped region) => 4^2= (4 * π*((4/2)^2)/2)  4 * (area of the each eyeshaped region) => 4= (π*(2^2)/2)  (area of the each eyeshaped region) => (area of the each eyeshaped region) = 2π 4
So finally, the area of the shaded region = (the area of the square)  2 * (each semicycle area) + (area of the each eyeshaped region) = 4^2 2 * π*((4/2)^2)/2 + (area of the each eyeshaped region) the area of the shaded region = 16 4π + (2π 4) = 12 2π
A. 12−π B. 12−2π > correct C. 12+π D. 12+2π E. 24−4π



Manager
Joined: 24 Jun 2019
Posts: 112

Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
Updated on: 23 Jul 2019, 20:50
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?See attached image below for visualization.Shaded Area = Area of Square  Area of 2 semicircles + Area of Overlap of Semicircles 1. SQUAREArea of Square = 4x4= 162. SEMICIRCLES:Semicircle hasdiameter 4, so it has radius 2. Area of 2 semi circles = \(Pi*2^2/2\) + \(Pi*2^2/2\) = 2Pi + 2Pi = 4Pi3. OVERLAP:The semicircles both have radius 2 and they will both pass through center of the square where they intersect. (See diagram below) So imagine the overlap as overlap of 2 Quarter circles. Now, isolate 1 quarter circle .... Here, Area of quarter circle  area of right triangle = half of the overlap Half Overlap = Area of Quarter Circle  Area of Right Triangle = \(Pi*2^2/4\)  1/2*2*2 = Pi  2 Area of Overlap = 2 x half overlap = 2Pi  4Area of Shaded Region = 16  4Pi + 2Pi  4 = 12  2PiANSWER B. \(12−2π\)Attachment:
GOT 23jul19 Geometry.jpeg [ 144.95 KiB  Viewed 1434 times ]
Originally posted by Vinit1 on 23 Jul 2019, 09:14.
Last edited by Vinit1 on 23 Jul 2019, 20:50, edited 1 time in total.



GMAT Club Legend
Joined: 18 Aug 2017
Posts: 5245
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)

Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 10:58
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
A. 12−π B. 12−2π C. 12+π D. 12+2π E. 24−4π
Area of square  (area of 2 semicircle  area of the common segment) = 16(4π(2π4) = 164π+2π4 = 122π
out of given options nearest value is IMO B 122pi
Originally posted by Archit3110 on 23 Jul 2019, 09:15.
Last edited by Archit3110 on 24 Jul 2019, 10:58, edited 1 time in total.



Manager
Joined: 08 Jan 2018
Posts: 145
Location: India
Concentration: Operations, General Management
WE: Project Management (Manufacturing)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:16
IMOBRefer attached image, Side of square ABCD = 4 units Now , divide the square in 4 parts by drawing lines (EF & GH) through midpoints of opposite sides.Area of shaded region 1= 1/4 * π * r^2 = 1/4 * π * 2^2 = π .... {quarter circle}Area of shaded region 2= 1/4 * π * r^2 = 1/4 * π * 2^2 = π .... {quarter circle}Area of shaded region 3 = 1/4 * 4^2= 4 .... {quarter of square}Area of region req. = Area of square  [ area of shaded region (1+2+3) ]Req. Area = 16  ( 4 + π + π ) = 12 2 π
Attachments
WhatsApp Image 20190723 at 9.33.54 PM.jpeg [ 66.06 KiB  Viewed 1971 times ]



Senior Manager
Joined: 27 Aug 2014
Posts: 356
Location: Netherlands
Concentration: Finance, Strategy
GPA: 3.9
WE: Analyst (Energy and Utilities)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:21
IMO answer is B:
Instead of doing long calculations, I tried to approximate the area. As both the circles are semicircles, they will intersect at the center of the square. so the top right part is slightly greater than 1/4th of the area of the square+ some more coming from the left over area after semicircles intersect 1/4th of the area of square is 4 units + approx. 1 unit from left semi circle + 1 unit from below semi circle = ~6units.
None of the answer choices have this value except B, approximating pi to 3.14.



Senior Manager
Joined: 16 Jan 2019
Posts: 498
Location: India
Concentration: General Management
WE: Sales (Other)

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:42
Since the two semi circles are congruent, they intersect at the top of the arc. We can divide this into 3 regions I. A quarter circle with radius 2 (Area = \(\frac{\pi*2*2}{4} = \pi\)) II. A square with side 2 (Area = \(2*2 = 4\)) III. Another quarter circle with radius 2 (Area = \(\frac{\pi*2*2}{4} = \pi\)) Total area = \(4*4 = 16\) Shaded area = Total Area  I  II  II Therefore shaded area = \(16  \pi  4  \pi = 122\pi\) Answer is (B)
Attachments
Untitled.png [ 9.44 KiB  Viewed 1900 times ]



Director
Joined: 24 Nov 2016
Posts: 739
Location: United States

Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 09:32
Quote: Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region?
A. 12−π B. 12−2π C. 12+π D. 12+2π E. 24−4π Sqr area=4ˆ2=16 Circle area=2ˆ2π=4π Semi area=2π Isosceles area=2 Segment area=Semi area/2Iso area=2π/22=π2 Shaded area = Sqr area  Circle area + 2 Segment area Shaded area = 16  4π + 2(π2) = 12  2π Answer (B).
Originally posted by exc4libur on 23 Jul 2019, 09:45.
Last edited by exc4libur on 24 Jul 2019, 09:32, edited 1 time in total.



Senior Manager
Joined: 10 Jan 2017
Posts: 329
Location: India

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 09:57
IMO correct answer is B  Explanation provided as attachment
Attachments
IMG_20190723_222230.JPG [ 1.08 MiB  Viewed 1881 times ]
_________________
Good, better, best. Never let it rest. 'Till your good is better and your better is best. Please hit +1 Kudos if you like my Post.



SVP
Joined: 03 Jun 2019
Posts: 1834
Location: India

Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
Show Tags
23 Jul 2019, 10:01
Two semicircles are drawn on adjacent sides of a square with side length 4 as shown above. What is the area of the shaded region? A. 12−π B. 12−2π C. 12+π D. 12+2π E. 24−4π Please see the solution in image below IMO B
Attachments
20190723_222756 2.jpg [ 1.06 MiB  Viewed 1864 times ]
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts." Please provide kudos if you like my post. Kudos encourage active discussions. My GMAT Resources:  Efficient LearningAll you need to know about GMAT quantTele: +911140396815 Mobile : +919910661622 Email : kinshook.chaturvedi@gmail.com




Re: Two semicircles are drawn on adjacent sides of a square with side len
[#permalink]
23 Jul 2019, 10:01



Go to page
1 2 3
Next
[ 60 posts ]



