What is the area of circle O ? (1) The ratio of the area of the squar

Question

What is the area of circle O ?

(1) The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
(2) The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.

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SiffyB · Answer

STATEMENT 1:  Let the radius of circle = r units
 Let the side of the inscribed square be a units.
Let's focus on the inscribed square first. 
The diagonal of this square will be the diameter of the circle. Using pythagoras algorithm, we have -
 (2r)^2 = a^2 + a^2  
 Upon solving, we get : a = \sqrt{2r}
For the circumbscribed square, the diameter of the circle = the side of the square.
Therefore, side of sqaure = 2r

The area of the square circumscribed about circle O ^ The area of the square inscribed in circle O
 = 2r^2 ^ sqrt(2)
 = 2 : 1

This is already stated in STATEMENT 1. The radius of the circle can be of any measurement. 
 Insufficient  (BCE)

STATEMENT 2:

As discussed in statement 1, the ratio of the length of the square circumscribed = 2r = diameter of the circle.
Thus, their ratio is 1:1.

Again, we already knew this. The value of radius can be anything for this rule to hold true.
 Insufficient  (CE)

Even upon combiing the two statements, we don't have a fixed value of the radius, thus,  (E).

bM22 · Answer

Let the radius of the circle be r.
Now the side of circumscribed square = 2r.
=> Area of the circumscribed square = 4r^2.

For the inscribed square, the radius of the circle would be the diagonal of the smaller square.
Thus the side of the inscribed square = \sqrt{r}
=> Area of the inscribed square = r.

Statement 1: The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
=> 4r^2/r = 2r.
We still dont know the value of r to find the area. Thus  Insufficient .

Statement 2: The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.
2r/2r = 1:1.
This information is of no use to us, as we cannot calculate the value of r, thus the area.
Thus,  Insufficient .

Statement 1+ Statement 2 : We have only 2r, but no value of r, thus we cannot calculate the area of the circle.
So  Insufficient .

Answer E.

anupam87 · Answer

Hi  Bunuel , this question asks the area of the circle. we know to calculate the ratio we will need to find the radius. 
Since, statement 1 and statement 2 both talks about ratio, I selected the answer E, because from the ratio there is no way to find a definite value for the radius. I did not try to solve the problem by analyzing each of the statements. Was my approach incorrect?

Basshead · Answer

Combined, the two statements are not sufficient. In each statement, we're simply given a ratio. From a ratio, we can't determine the area of the circle. The ratios we're given are true for any circle and its circumscribed and inscribed squares.

Answer is E.

Attachments

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Answer is E.

GMATWhizTeam · Answer

Step 1: Analyse Question Stem

Area of a circle = π * \(r^2\), where r is the radius of the circle.
To find the area of circle O, we need information about its radius or diameter.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
When a square is circumscribed about a circle – in other words, when a circle is inscribed in a square – the diameter of the circle is equal to the length of the side of the square.

Attachment:

31st May 2022 - Post 3 - 1.png [ 6.23 KiB | Viewed 5359 times ]

If the radius of circle O is r, the length of the side of the circumscribing square = 2r.
Therefore, area of square = \((2r)^2\) = 4\(r^2\)

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

Attachment:

31st May 2022 - Post 3 - 2.png [ 6.43 KiB | Viewed 5334 times ]

Since the radius of circle O is r, diameter = 2r; therefore, length of diagonal of square inscribed in circle O = 2r.

Area of a square = ½ * \((length of diagonal)^2\).

Therefore, area of inscribed square = ½ * \((2r)^2\) = ½ * 4\(r^2\) = 2\(r^2\).

The ratio of the areas of the two squares = 2 : 1. This is a fact that can be proven and Statement 1 states the same fact.
Knowing this fact will not help us find out the radius of the circle.

The data in statement 1 is insufficient to find a unique value of the area of circle O.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.
When a square is circumscribed about a circle – in other words, when a circle is inscribed in a square – the diameter of the circle is equal to the length of the side of the square.

Attachment:

31st May 2022 - Post 3 - 1.png [ 6.23 KiB | Viewed 5359 times ]

If the radius of circle O is r, the length of the side of the circumscribing square = 2r.

Again, statement 2 is stating a mathematical fact. This is not sufficient to find the radius of circle O.

The data in statement 2 is insufficient to find a unique value of the area of circle O.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
From statement 2: The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.

Both statements state already known facts, but do not provide us with the value for the radius. Circle O could be a very small circle or an extremely large one.
Even after combining the information from both statements, we do not have a unique figure and hence will not have a unique value for the area.

The combination of statements is insufficient to find a unique value of the area of circle O.
Statements 1 and 2 together are insufficient. Answer option C can be eliminated.

The correct answer option is E.

bumpbot · Answer

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What is the area of circle O ? (1) The ratio of the area of the squar

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What is the area of circle O ? (1) The ratio of the area of the squar

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