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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 http://en.wikipedia.org/wiki/Isoperimetric_inequality.

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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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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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