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

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Re: What is the area of circle O ? (1) The ratio of the area of the squar [#permalink]
Bunuel wrote:
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.

DS21214

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?
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Re: What is the area of circle O ? (1) The ratio of the area of the squar [#permalink]
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.

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Re: What is the area of circle O ? (1) The ratio of the area of the squar [#permalink]
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 4158 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 4130 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 4158 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.
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Re: What is the area of circle O ? (1) The ratio of the area of the squar [#permalink]
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Re: What is the area of circle O ? (1) The ratio of the area of the squar [#permalink]
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