An artist wants to place a piece of round stained glass into a square that is made of copper wire

stained-glass
copper-square

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An artist wants to place a piece of round stained glass into a square that is made of copper wire. See the figure. If the perimeter of the square is 40 inches, what is the area of the largest circular piece of stained glass that can fit into the copper square?

Answer:

The answer would be 25pi. Simplified, it would be: 78.53981633974483

Since all four sides of a square are equal length, we can say that the perimeter = 4s, where s is the length of one side.

We know that the perimeter is 40 inches, so we can say that 40 = 4s, or s = 10 inches.

The width of the circle (aka the diameter) has to be equivalent to the width of the square in order to fit properly. Therefore, the radius of the circle is exactly half the width of the square.

Thus, the radius of the circle is 10/2 = 5 inches.

The rest is simple: plug in r = 5 into the formula for the area of a circle (A = pi * r^2) and obtain your answer:

25 pi.