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

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?

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.

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.



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.



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.



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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