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If a string with length of 20 cm was to create a rectangle and circle, would area of these objects be the same?

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No. The circle will always have larger area than the rectangle. See

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No, the circle is the shape with the largest area for its perimeter. From $C=2\pi r$ we find $r=\frac{10}{\pi}$ and $A=\pi r^2=\frac {100}{\pi} \approx 31.83 \text{cm}^2$. This is larger than both rectangles mentioned by Chris Card. A square would have area $25\text{cm}^2$

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In this case, the circle has an area $\approx 21.46\%$ greater than that of the square. – Nick Sep 23 '14 at 11:37

No. You can even create two rectangles with different areas but the same perimeter, e.g. sides 4 and 6 (area 24) and sides 1 and 9 (area 9).

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Since the square is the optimal area of any rectangle with fixed perimeter, it is reducible to comparing area of square to area of circle.

Let square's perimeter $=$ circle's circumference $=2\pi r,$

then area of circle is $\pi r^2$ and area of square is $\left(\dfrac{2\pi r}{4}\right)^2.$ The former is of course greater than the latter, since $\pi>\dfrac{\pi^2}{4}.$

A plot of the two to help visualise it (square area: orange, circle area: blue):

enter image description here

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