GMAT Club Around The World!!! A circle is inscribed in a square. What

Question

https://gmatclub.com/forum/download/file.php?id=59264 A circle is inscribed in a square. What is the area of the shaded region? (1) The distance from the center of the circle to a vertex of the square is 2\sqrt{2} (2) The square has a perimeter of 16. Happy New Year Samoa and Christmas Island/Kiribati ! Dec 31 Event: GMAT Club Around The World ! (38 new questions posted every hour time zone on Dec 31!) https://gmatclub.com/forum/download/file.php?id=59744 images.png

Deepakjhamb · Answer

A circle is inscribed in a square. What is the area of the shaded region?

(1) The distance from the center of the circle to a vertex of the square is 2√2
(2) The square has a perimeter of 16.

Solution :

Lets analyze the stem :
let a be side of square 
shaded region area = a^2 ( 1- pie /4) , so we only need to find a

statement 1: diagonal /2 = 2underroot 2 so we can definitely find a as d = underoot 2 * a

so sufficient

statement 2: 4a = 16 so we can find a

so sufficient

so answer is d) : each statement alone is sufficient to answer the question

Sagart21 · Answer

1) distance from center of circle to corner of the square is given

This forms a triangle with other 2 sides being Radius of the circle and half of the side of square 
As it is a 45-45-90 triangle
Other 2 sides are 2 units each.

Hence Area of circle and square can be found and then the area of shaded region.

1 is sufficient.

2) from perimeter side is 4 and half of the side is 2
Area of square can be found.

Diameter of circle is equal to the side of the square and Radius is half of the side = 2

Hence area of circle can be found.

So area of shaded region can be found.

2 is also sufficient.

Both statements are sufficient individually.

Answer D

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bidskamikaze · Answer

A circle is inscribed in a square. What is the area of the shaded region?

(1) The distance from the center of the circle to a vertex of the square is 2√2 
This is sufficient for finding the side of square (=4), which is equal to the diameter of the circle.
Area of square - Area of circle = Area shaded region.
Sufficient.

(2) The square has a perimeter of 16. 
This is sufficient for finding the side of square (=4), which is equal to the diameter of the circle.
Area of square - Area of circle = Area shaded region.
Sufficient.

Answer D.

rocky620 · Answer

Let O be the center of the circle of radius  r , and ABCD is the square of side  x

r=x/2

Area of shaded region =  x^2  - pi* x^2 /4

We need to find x

(1) The distance from the center of the circle to a vertex of the square is 2√2 
OA = 2√2, so the diagonal = 4√2
 x=4 
  Sufficient

(2) The square has a perimeter of 16.
 x=4 
  Sufficient

IMO Option D

BIDJI23 · Answer

If we know the diameter of the circle, then we know the length of the side of the square. Or if we know the length of a side, we know the diameter of the circle. And then it's side*side of the square - minus the area of the circle.

1 is sufficient.
2 is sufficient

PUSH D

1 is sufficient. 
2 is sufficient

PUSH D

anjanita · Answer

When a circle is inscribed in a square, the length of each side of the square is equal to the diameter of the circle.

The first statement says that the radius of the inscribed circle is 2√2 units and therefore the area of a circle of radius r units is A=πr2 . Substitute r=2√2 in the formula. And the area of the square will be S=(2*2√2)^2

So, the area of the shaded region is S-A...therefore statement 1 is sufficient
 
Statement 2 says the perimeter of the Square is 16. therefore, the side of the square is 4, there fore area S = 4^2
The area of the circle is = π*(4/2)^2

Therefore the area of the shaded region is S-A...... therefore statement 2 is sufficient

Therefore the answer is D

unraveled · Answer

https://gmatclub.com/forum/download/file.php?id=59264 
A circle is inscribed in a square. What is the area of the shaded region?

As circle is inscribed in the square, knowing any parameter - be it radius/diameter/perimeter/area of circle OR length of side/diagonal/perimeter, area of square - of either the square or the circle would be sufficient to get the answer. 
2*radius of circle(r) = length of side of square(s)
Thus, any relation, if given, is sufficient.

(1) The distance from the center of the circle to a vertex of the square is  2\sqrt{2} 
 2\sqrt{2} = \sqrt{2}*\frac{s}{2}

SUFFICIENT.

(2) The square has a perimeter of 16.
4s = 16

SUFFICIENT.

Answer D.

ananya3 · Answer

IMO D

(1) The distance from the center of the circle to a vertex of the square is 2√2
distance from center of circle to vertex = diagonal of square/2
=> diagonal of square = 4√2
=> sides of square = 4 (diagonal = a√2; where a is the side of a square)
=> diameter of circle = 4 = radius * 2
=> area of square = 4*4 = 16
area of circle = π*2^2
=> area of shaded region = 16 - 4π

(2) The square has a perimeter of 16.
=> sides of square = perimeter/4 = 4
=> diameter of circle = 4 = radius * 2
=> area of square = 4*4 = 16
area of circle = π*2^2
=> area of shaded region = 16 - 4π

Harsh9676 · Answer

IMO D

We need to find the area of the shaded region.

So Area of shaded region = Area of Square - Area of the square
 
Area of shaded region = Pi r^2 - S^2  .............................(1)

Diameter of the Square = Side of the Square; So if we know the radius or the side of the square we can find the area of the shaded region.

St1: The distance from centre of the circle to the vertex of the square = 1/2 (Diagonal of the square s(sqrt(2); from this, we can get the side and hence the area of shaded region.

St 1 is Sufficient
 
St2: The square has a peimeter of 16.

So the side = 4 and again we can get the area of the shaded region.

St2 alone is also sufficient.

bobnil · Answer

concept: 
Area of a circle inscribed to square having the length of a =  \pi/4*a^2 
Now, area of shaded region = area of the square - the area of the inscribed circle
 = a^2-\pi/4*a^2 
now if we come to know about the value of  a  or, radius of circle  ( = a/2)  then it will be sufficient.

Statement I :  1/2  of the diagonal given ... so value of  a  can be found out. Sufficient..
Statement II: perimeter of square is given. So value of  a  can be found out. Sufficient..

So both statements are sufficient individually
Ans D

MHIKER · Answer

(1) It's is half the diagonal of the square, so the diagonal of the square = 2\sqrt{2}+2\sqrt{2}=4\sqrt{2} .

We know,  d=s\sqrt{2} ; d is a diagonal and s is a side of the square.

Thus,  s\sqrt{2}=4\sqrt{2}

s=4 , which is the meter of the circle; sufficient.

(2)  4s=16, \ s=4 , which is also the diameter of the circle. Sufficient.

The answer is D.

TarunKumar1234 · Answer

Stat1: The distance from the center of the circle to a vertex of the square is 2√2
Half of the diagonal = a/√2 = 2√2, a= 4= side of square= Diameter of Circle
Grey area= 4^2 - Pir * 2^2. Sufficient

Stat2: The square has a perimeter of 16.
side of square= 16/4 = 4 = Diameter of Circle
Grey area= 4^2 - Pir * 2^2. Sufficient

So, It is D.

jhavyom · Answer

https://gmatclub.com/forum/download/file.php?id=59264 
A circle is inscribed in a square. What is the area of the shaded region?

(1) The distance from the center of the circle to a vertex of the square is  2\sqrt{2} 
(2) The square has a perimeter of 16.

For a circle with radius r, the side of the square will be 2r.
Required area = (2r)^2 - pi*r^2
So, if radius can be determined, the required area can be determined.

(1) The distance from the center of the circle to a vertex of the square is  2\sqrt{2} 
So, diagonal of square =  4\sqrt{2} 
or side of square = 4
So, Radius = 2

Sufficient

(2) The square has a perimeter of 16.
Side of square = 4
So, Radius = 2

Sufficient

Choice D is the answer.

mdsaddamforgmat · Answer

IMO: D
St1- from the Given distance we can calculate the area of both Circle & square and subtracting both the areas will give the shaded region are.
Sufficient
St2: 
From the given perimeter, we can calculate the square's side length and consequently the area of both Square & circle and hence, the shaded area…
Sufficient.

RajatJ79 · Answer

Solution -

Statement (1) alone -
 (1) The distance from the center of the circle to a vertex of the square is  2\sqrt{2}  
This is distance is half the length of the diagonal of the square. Which means diagonal length is 4√2.
Length of side of square is this 4.
 Statement (1) alone is sufficient.

Statement (2) alone -
 (2) The square has a perimeter of 16. 
Using the length of side of square area of share and circle can be found out.
Thus, area of shaded region can be calculated.
 Statement (1) alone is sufficient.

Answer should be Option D.

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aditikamra · Answer

Hello,

Answer should be option B only can help us derive the area of the shaded region

Green2k1 · Answer

Correct answer D

A circle is inscribed in a square. What is the area of the shaded region?

Shaded region area = area of square - area of circle
 = a^2 - pi*(a/2)^2
 
(1) The distance from the center of the circle to a vertex of the square is 22√22
  - can be calculated using Pythagoras theorem
= (a/2)^2 +(a/2)^2 = (22sqroot22)^2
-Sufficient

(2) The square has a perimeter of 16.
 = 4a = 16
- sufficient

Neha2404 · Answer

Imo-D

1) assume center of circle is O
 Radius on tangent create angle of 90
45:45:90
2:2:2rt2
Radius is =2
Side of square =4
Shaded area is =area of square- area of circle
Sufficient

statement-2
Perimeter of square =16
Side of square =4
Radius of circle is =2

Sufficient

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manasi05 · Answer

To find the area of the shaded region we will need the value for area of square and area of circle.
Area of the shaded region = Area of Square - Area of Circle

Statement 1:
This tells us the distance from the center of the circle to a vertex of the square 2√2. So twice of 2√2 will be the diagonal of the square. 
So diagonal of square = 4√2
Using the formula D= s√2 we will get the side of square to be 4
So area of square = 16
Since the circle is inscribed inside the square so the value for diameter of the circle will be the side of the square.
So Diameter =4 and Radius= 2
Area of circle= π r^2 = 4π
Area of Shaded region = 16-4π
Sufficient

Statement 2
The perimeter of square is 16 
Using the perimeter formula we can find the side of the square which will be 4 (Perimeter of square= 4s)
So area of square = 16
Since the circle is inscribed inside the square so the value for diameter of the circle will be the side of the square.
So Diameter =4 and Radius= 2
Area of circle= π r^2 = 4π
Area of Shaded region = 16-4π
Sufficient

Hence the answer will be D

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