A square is there as above figure such that its one side’s length is 8

Question

The side of a square has the length of 8. What is the area of the region shaded? 
 A. 48-8π 
 B. 48-6 π 
C. 24+6π 
D. 16+8π 
 E. 64-8π

*An answer will be posted in 2 days.

HMC · Accepted Answer

Area of Shaded region = Area of Square - 2* area of half circle + Area of that "petal of flower" ....(1)

The critical part here is knowing area of "Petal of flower"...

Assume that you draw four semicircles within square ...

so 8^2 = 4*area of half circle - 4*Area of that "petal of flower"

FROM ABOVE you can derive are of petal of flower as 8pi-16 ... and then put that value in (1) above....

Hoping to get first Kudo on the forum....

paidlukkha · Answer

area of square = 8^2 = 64
IMO E (only one option for area of square - area of unshaded region)

Why is OA as A?

rohit8865 · Answer

No. E is not the correct choice
option E is the area of square-area of one semicircle--------(8*8)-pi*(4^2)/2
but we need area of 2 semicircle -area of combined spaces.

logically answer choice will be a value lesser than E

paidlukkha · Answer

Keeping in mind ways to tackle probability, I'm considering whole - not required = required in this case, area of square - shaded = unshaded region area of square = 64 only one option! Sorry, but I still do't understand

paidlukkha · Answer

MathRevolution  Source of this question?

If official, waiting for official solution

bb   Bunuel

Bunuel · Answer

You can check the source in the tags. It's Math Revolution.

Why2settleForLess · Answer

Approach 1:

shaded region= Area of square - ((2 x Area of semicircle) - Area of region swept two times by semicircles)

Now Let's look at the area swept twice by the semicircles,

Area of region swept two times by semicircles = (2 x Area swept by quater circle) - Area of that small square of side 4
Area of region swept two times by semicircles = (2 x Pi x 4 x 4 /4) - (4 x 4)
Area of region swept two times by semicircles = 8Pi-16

Now coming back to the original equation,
shaded region= Area of square - ((2 x Area of semicircle) - Area of region swept two times by semicircles)
shaded region= Area of square - (2 x Area of semicircle) + Area of region swept two times by semicircles
shaded region= 64 - (2x Pi x 4 x 4/2) +8Pi -16
shaded region= 64 - 16Pi +8Pi -16
shaded region= 64 - 16Pi +8Pi -16
shaded region= 48 - 8Pi

Approach 2: Ball Parking

since Figure is not as per scale, It may mislead you. Drawing figure nearly to the scale will give you a hint that the intersection of two semicircle is at the center of the square and if you draw a diagonal, you will get to know that the area of shaded region should be less than half of the area of square i.e., <32 (<64/2). Approximate the value of Pi to "a little more than 3". Only A and B satisfies the condition of area less than 32.
A will give you value "little less than 24". and B will give you value "little less than 30".
A tough call now. Now, accuracy depends on how nearly the sketch you have redrawn on your scratch pad is to the scale.
you will realise that B (little less than 30) is too near to 32 to be true.
So, Go with A.

Approach 2 is advisable only if you are not getting any other method on your mind.

Kudos if the answer deserves one.

Attachments

puzz.JPG [ 37.43 KiB | Viewed 5949 times ]

DigitsnLetters · Answer

I guess it should be " + Intersection area of the semi-circles " (i.e. area of the "petal" as you said) coz this area was subtracted twice in the expression "2*area of half circle".

But kudos nonetheless.

HMC · Answer

Rahul you are correct... This is how I end up making silly mistakes ... Edited original post to correct the expression.

MathRevolution · Answer

The area of a square whose side is 8 – (the area of a square whose side is 4 + the area of the semi-circle whose side is 4) = the area of the region shaded
The correct answer is A.

hatemnag · Answer

Actually, I can not understand the solution above. 
Please experts help me in this question step by step

PKay · Answer

Ref the figure:
 Required area is  1+2+3

1 ----- it is a square of side 4 units. area=16

2 -----area of square of side 4 units - R4-1

3 -----area of square of side 4 units - R5-2

R4-1  is nothing but 1/2 of  (4)  which itself is a semi-circle with radius 4 units.
=> value of  R4-1  is nothing but 1/4 of circle of radius 4 units-16pi/4= 4pi.

So is  R5-2 = 4pi.

So, requisite area is  1+2+3 
16+(16-4pi)+(16-4pi)

=48-8pi 
A is the OA.

+1 Kudos if it helps.

sjavvadi · Answer

Good Question. I like the "area of petal approach" from HMC.

hatemnag · Answer

Great. you smartly structured the answer and became very organized to me. this is exactly what I missed. the approach by which I quickly answer the question within 2 minutes. It looks complicated one but by approach it became reachable. The geometrical concepts required are simple.
sure I gave you Kudos
thank you

Fdambro294 · Answer

We have 2 semicircles with radius 4 that will come up and intersect in the center of the Square (the figure isn’t exactly drawn to scale).

This Center of the Square Point will be the point that is right above the midpoint of the Left Vertical Side of the Square and right above the midpoint of the Bottom Horizontal Side of the Square - call this center of the square point P

Midpoint on Left Vertical Side of square = Center of the semi-circle = A

Midpoint on the Bottom Horizontal Side of the Square = center of the semi-circle = B

Draw perpendicular radii ——-> PA = PB = (1/2) (diameter of semi-circle) = (1/2) (8) = 4

The 2 radii, PA and PB, are adjacent sides of a 4 by 4 square in the bottom left portion of the square.

Since these 2 radii have cut both semi-circles in half, the area that is outside this 4 by 4 square but inside each of the semi-circles is (1/2) of each of the semi-circles or (1/4) of an entire circle with radius 4

Therefore, starting from the Upper Left Vertex of the Square and moving down along the bottom side:

-we have (1/4) of a circle with radius 4 up until radii PA

-then we have a Square that is 4 by 4

-and finally we have 1/4 of a circle with radius 4 beyond radii PB

These 3 portions represent the area that is NOT Shaded in the Square.

Subtract the area of the 3 portions from the total area of the square and you will get the shaded area.

(8)^2 -

64 -

48 - (8) (pi)

Answer A

Posted from my mobile device

Fdambro294 · Answer

Posted above is the step by step solution to find this answer.

Posted from my mobile device

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

A square is there as above figure such that its one side’s length is 8

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A square is there as above figure such that its one side’s length is 8

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards