what is the area of the circumscribed square about a circle having a 12pie perimeter?

Doesn't the side of the square equal the diameter of the circle ?

so let's find the radius,
12pi = dpi
d = 12
area of square = 12 x 12 = 144

To find the area of the circumscribed square around a circle, we need to know the radius of the circle.

The perimeter of a circle is given by the formula: perimeter = 2πr, where r is the radius.

In this case, the perimeter is given as 12π. So we can set up the equation: 2πr = 12π.

Now let's solve for the radius. Divide both sides of the equation by 2π: r = 6.

The radius of the circle is 6 units.

In a circumscribed square around a circle, the diagonal of the square is equal to the diameter of the circle, which is twice the radius. Therefore, the diagonal of the square is 2r = 2 * 6 = 12 units.

The area of a square can be calculated using the formula: area = side length^2.

To find the side length of the square, we can use the Pythagorean theorem. The diagonal of the square (12 units) is the hypotenuse, and the sides of the square are the legs of the right triangle formed.

Using the Pythagorean theorem: side length^2 + side length^2 = diagonal^2.
So, side length^2 + side length^2 = 12^2.

Simplifying the equation: 2 * side length^2 = 144.

Divide both sides by 2: side length^2 = 72.

Taking the square root of both sides: side length = √72.

Therefore, the side length of the square is √72 units.

To find the area of the square, we apply the formula: area = side length^2.

Substituting the side length: area = (√72)^2 = 72 units^2.

Therefore, the area of the circumscribed square about a circle with a 12π perimeter is 72 square units.