What is the area of the square, if four vertices lie on the circumference of a circle where the area of thecircle is four times its diameter in magnitude?(a) 28 sq.units (b) 216 sq.units (c) 232 sq.units (d) 264 sq.units (e) 2128 sq.un

To find the area of the square, we need to follow these steps:

Step 1: Determine the diameter of the circle. Given that the area of the circle is four times its diameter, we can use the formula for the area of a circle, A = πr^2, where r is the radius. Since the area is four times the diameter, we can write the equation as 4d = πr^2, where d is the diameter. Solving for d, we get d = 2r.

Step 2: Find the length of the side of the square. Since the four vertices of the square lie on the circumference of the circle, the diagonal of the square is equal to the diameter of the circle. Therefore, the length of the side of the square is equal to the radius of the circle, which is half the diameter. So, the length of the side of the square is r.

Step 3: Calculate the area of the square. The area of a square is given by the formula A = s^2, where s is the length of a side. In this case, the length of the side is r, so the area of the square is A = r^2.

Step 4: Substitute the value of r. Since we know that d = 2r, we can substitute this value into the equation for the area of the circle to find r. 4d = πr^2 becomes 4(2r) = πr^2, which simplifies to 8r = πr^2. Dividing both sides by r, we get 8 = πr. Solving for r, we find r = 8/π.

Step 5: Calculate the area of the square. Now that we have the value of r, we can substitute it into the equation for the area of the square. A = (8/π)^2 = 64/π^2.

Step 6: Simplify the area. To simplify the area, we can multiply the numerator and denominator by π^2 to get rid of the fraction. This gives us A = (64/π^2) * (π^2/π^2) = 64π^2/π^2 = 64.

Therefore, the area of the square is 64 square units.