If a square is inscribed in a circle that, in turn, is inscribed in a

Question

If a square is inscribed in a circle that, in turn, is inscribed in a larger square, what is the ratio of the perimeter of the larger square to that of the smaller square?

A.  \frac{1}{2}  
B.  \frac{1}{\sqrt{2}}  
C.  \sqrt{2}  
D.  2  
E.  3.14

M17-23

Ok, I understand this explanation from sportyrizwan:

Let "a" be the diameter of the circle 
Therefore the side of the bigger square will be a and the perimeter 4a

Also a will be the diagonal of the smaller square 
Therefore the side of the smaller square will be a/sqrt2 and perimeter (2a)sqrt2

Therefore the ratio of the bigger square to the smaller square is sqrt2

https://gmatclub.com/forum/square-inscribed-in-a-circle-inscribed-in-a-square-55210.html

***********

However, when I try to plug values, it doesn't work..

Let say larger square
one side x=2, so P=8

Smaller square
Diagonal = hypotenus = 2
So x^2+x^2=2^2
2x^2=4
x^2=2
x=sqrt2
P=4sqrt2

L/S = 8/4sqrt2 = 2 ???? It doesn't work and I can't find my mistake..

PLEASE help me to find my mistake!!

Thank you very much for your help!!

Bunuel · Accepted Answer

Look at the diagram: SquareCircleSquare.jpg You can see that a diameter of the circle equals to a diagonal of the smaller square and a side of the bigger square: diameter=diagonal_{small}=side_{big} ; Now, if the side of the bigger square is a then its perimeter will be P=4a ; Next, a side of a smaller square will be \frac{a}{\sqrt{2}} and its perimeter p=\frac{4a}{\sqrt{2}} ; \frac{P}{p}=\sqrt{2} Answer: C. Or as soon as you realize that the sides of bigger and smaller squares are a and \frac{a}{\sqrt{2}} respectively, so their ration is \sqrt{2} , then as P=4*side then the ratio of the sides will be the same as the ratio of the perimeters, so the same \sqrt{2} .

gettinit · Answer

I think your math is off when you are trying to find a side of the smaller square.

So Large square side =2. So diameter of circle =2 - therefore 2=hypotenuse of smaller square.

Since the square is two isosceles triangles - hypotenuse=x^2 + x^2 or in other words in our case 2/sqrt2 to get one of the sides.

so perimeter of larger square = 2*4= 8 perimeter of smaller square = 2/sqrt2 * 4 = 8/sqrt2.

when you then take the ratio of larger square perimeter to smaller square it looks like this:

8/(8/sqrt2) or 8 * sqrt2/8 so the 8's cancel out and your are left with sqrt 2 - the answer.

hope it helps.

KarishmaB · Answer

You got  \frac{L}{S} = \frac{8}{4\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}

Sometimes, it is the silliest things that get you!

maive · Answer

I got it!!

Thanks a lot for your help, I really appreciate it!!

Bunuel · Answer

Consider the diagram below: m17-23.png The side of a large square is a , thus its perimeter is 4a ; The side of a small square is \sqrt{(\frac{a}{2})^2+(\frac{a}{2})^2}=\frac{a}{\sqrt{2}} , thus its perimeter is \frac{4a}{\sqrt{2}} ; Hence the ratio is \frac{(4a)}{(\frac{4a}{\sqrt{2}})}=\sqrt{2} . Answer: C

Vishvesh88 · Answer

Bunuel  : Need your help here. Why is the diagonal of the small square = a/sqrt 2?

Bunuel · Answer

The diagonal of small square is a, not \frac{a}{\sqrt{2}} . \frac{a}{\sqrt{2}} is side of a small square. Check here: if-a-square-is-inscribed-in-a-circle-that-in-turn-is-inscribed-in-a-105993.html#p1419806

Vishvesh88 · Answer

Bunuel  : Got it mate! thanks.

narendran1990 · Answer

Assume the side of the larger square to be 8, perimeter of larger square = 32.

Diameter of the circle inscribed within a square = Side of the square , therefore radius = 4. Furthermore radius of the circle is half the diagonal length of smaller square, hence 4= √2a/2 , solving for 'a', we get 4√2. Therefore, perimeter of smaller square = 16√2. Finally, 32/16√2 = √2.

sashiim20 · Answer

Let the side of small square be  = 2

Therefore the diagonal of small square  = 2\sqrt{2}

Diagonal of small square is  =  diameter of the circle

Diameter of the circle is  =  side of the large square

Therefore side of large square  = 2\sqrt{2}

Perimeter of small square  = 4a = 4*2 = 8

Perimeter of large square  = 4a = 4*2\sqrt{2} = 8\sqrt{2}

Ratio of perimeter of large square to perimeter of small square  = \frac{8\sqrt{2}}{8} = \sqrt{2}

Answer C

LoneSurvivor · Answer

1. If a square is inscribed in a circle which is inscribed in a square then outer square area will be twice that of inner square area.
2. If a circle in inscribed in a square which is inscribed in a circle then the outer circle area will be twice that of inner circle area.

Paras96 · Answer

Let the radius of circle be "r"

For smaller square Let side be "a"
Since smaller square is inscribed in circle
in this case the diameter of the circle will be equal to diagonal of smaller square
2r=a√2------1

For larger square let side be "b"
Since circle is inscribed in larger square
in this case diameter of the circle will be equal to side of larger square

2r=2b-------2

Using 1 & 2

b=a√2

therefore ratio of perimeter will be equal to ratio of side = √2

Hence C

If a square is inscribed in a circle that, in turn, is inscribed in a

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

If a square is inscribed in a circle that, in turn, is inscribed in a

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