Awire 340 in. long is cut into two pieces. one piece is formed into a square and the other into a circle. if the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

The piece of wire cut for the square is 180.3 inches, and the piece of wire cut for the circle is 159.7 inches.
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Explanation:
Use the following formulas for the areas of a square and circle:

Square


Circle


s represents the length of the side in a square, and r represents the radius of the circle.

Set these two equations to be equal to each other.



We can simplify this by square rooting both sides to get s by itself:



Now we'll find the perimeter of the square and the circumference of the circle. Use the following formulas for the perimeter and circumference:

Square


Circle


The perimeter and circumference must both equal 340, so set the equations to add together to become 340:



Because we know what s equals, we can plug it into this equation:



Simplify the equation by dividing both sides by 2 and extracting r from the first term.



Assuming pi = 3.1416, replace pi with this decimal value and solve:













Rounded to the nearest tenth, the radius of the circle is 25.4.

We can now use the radius to find the circumference of the circle:





Rounded to the nearest tenth, the circumference of the circle is 159.7 inches.

Plug this value into the original equation for the perimeter and circumference:



Subtract 159.7 from both sides.



The perimeter of the square is 180.3 inches.

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