If the circumference of a circle is the same as the perimeter of the square with a side as 4 pi in, what is the area of the circle in square inches?
since c = 2πr, r = c/2π
since a circle has a = πr^2, a = π(c/2π)^2 = c^2/4π
the square has perimeter 4*4π = 16π
so, the circle has area
a = (16π)^2/4π = 64π
To find the area of a circle, we first need to determine the radius of the circle.
Let's start by finding the perimeter of the square. The formula for the perimeter of a square is given by P = 4s, where P represents the perimeter and s represents the length of one side. In this case, the side length of the square is 4π inches, so the perimeter of the square is 4 * 4π = 16π inches.
Now, let's set up an equation related to the circumference of the circle and the perimeter of the square. The circumference of a circle is given by the formula C = 2πr, where C represents the circumference and r represents the radius. Since the given circumference is equal to the perimeter of the square, we can write:
2πr = 16π
Dividing both sides of the equation by 2π, we get:
r = 8
Now that we have determined the radius of the circle, we can find its area. The formula for the area of a circle is given by A = πr^2, where A represents the area and r represents the radius. Substituting the value of the radius into the formula, we have:
A = π * 8^2
= π * 64
= 64π square inches
Therefore, the area of the circle is 64π square inches.