Square ABCD is inscribed in circle O. What is the area of square regio

1.
Area of circle = pi*r^2 = 64 pi
=> r= 8
Diameter = 2*8= 16
Diagonal of the square is equal to diameter of the circle.
Now applying pythagorean theorem ,
Let s = side length
(s)^2 + (s)^2 = (diagonal)^2
=> 2 *(s)^2 = 256
=>(s)^2 =128

Sufficient.

2 . 2*pi*r= 16 * pi
=> r = 8
We can calculate the area as in 1 .
Sufficient.

Answer D
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Given: A square is inscribed in a circle. Therefore, the Diameter of circle = Diagonal of the square
Area of \(square = \frac{diagonal^2}{2}\)

Hence, all we need to know is the diameter of the circle.

S1- Given the area of circle, We can obtain the radius and hence, the diameter. - Sufficient
S2- Given the circumference, like S1 we can obtain diameter. - Sufficient

Both statements alone are sufficient- D

Area of circumscribed circle = (pi/2)*Area of square
Area of square = (2/pi)*Area of circumscribed circle.
So St1 is sufficient.

St2: 2*pi*r = 16*pi --> r = 8... Area of circle can be found out.
St2: Sufficient.

Answer: D

Once you find the diameter [d = 16] of circle, you can find the side [s] of a square using the relation:

d = s * sq.root(2)

s = 16/sq.root(2)

Answer is D as the diameter of circle is the diagonal of the square.

Since, square ABCD is inscribed in the circle, the only way that is possible is when the diagonal of the square is equal to the diameter of the circle.
AB\(\sqrt{2}\) = diameter of circle = 2r

1) Area of O is 64π, i.e. \(πr^2\) = 64π.
From this we can find the radius of the circle, thus determine the diameter and thereby we can deduce the area of the square ABCD.

2) Circumference of O is 16π, i.e. 2πr = 16π.
From this we deduce radius, r and finally the area of square ABCD.

Using Formula :-

Area of a circumscribed circle = 1.57 * Area of the square

St 1 - Sufficient
St 2 - Sufficient as C of cricle we can apply the above formula easily by calculating the area of circle

Answer D

Draw the circle and inscribe a square to see that: The radius of the circle is half of the square's diagonal and hence all you need to to know is the radius from the circle in order to calculate the area of the square.

Area of a square can be calculated as \(\frac{D^2}{2}\)

Statement 1: From this we can calculate the radius since r^2π=64π, therefore r = 8
Statement 2: From this we can too, calculate the radius 2rπ = 16π, therefore 2r=16, r = 8

From both statements we can calculate the diagonal = 16 (=2*r) and hence the area of the square.

Answer D.

There is nothing to solve here.
We know ABCD is inscribed so its vertices are on the circle and the diagonal is the diameter of the circle.
Once we know the diameter of circle we can find the area = D1*D2/2. here diagonal 1=diagonal 2 because both are diameter of circle and diagonal of a square.
1. we can find the diameter from area.
2. we can find the diameter from circumference.

So Answer is D

The area of any square is also given by the formula \(\frac{diagonal^2}{2}\)
In a square inscribed inside a circle the diagonal becomes equal to the diameter
SO if we can figure out the radius we can double it to find the diameter and hence the diagonal of the square and finally the area

Option 1 ) gives us the area ; radius can be calculated easily ; SUFFICIENT
Option 2) gives us the circumference ; again radius be calculated easily; SUFFICIENT

ANSWER IS D
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The area of any square is also given by the formula \(\frac{(diagonal^2)}{2}\)
In a square inscribed inside a circle, the diagonal of square is equal to the diameter of the circle
SO if we can figure out the radius we can double it to find the diameter and using diameter we can figure the area of Square by using \(\frac{(diagonal^2)}{2}\)

Option 1 ) gives us the area ; radius can be calculated easily ; SUFFICIENT
Option 2) gives us the circumference ; again radius be calculated easily; SUFFICIENT

ANSWER IS D
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Thank you for the explanation. It helps.

I also wanted to know if Square inside a circle with corners touching the circle can also be an equilateral triangle?
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Bunuel, why should we assume the corners of the square touch the circle? It just says the square is inscribed, nothing about how the inscription is made. Maybe I'm just overthinking the wording, but it really bothers me it doesn't specifically say "Sqaure ABCD inscribed in circle O with corners of sqaure touching the outside of the circle.

In geometry "inscribed" means drawing one shape inside another so that it just touches. It does not mean just inside.
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Answer: Option D

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Solution:

Question Stem Analysis:

We need to determine the area of square region ABCD, which is inscribed in circle O. Recall that the diameter of a circle is the diagonal of its inscribed square. Therefore, if we can determine the radius (and hence the diameter) of the circle, then we can determine the diagonal (and hence the side and area) of the inscribed square.

Statement One Alone:

Since we are given the area of the circle, we can determine its radius, and hence we can determine the area of the inscribed square. Statement one alone is sufficient.

Statement Two Alone:

Since we are given the circumference of the circle, we can determine its radius, and hence we can determine the area of the inscribed square. Statement two alone is sufficient.

Answer: D
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