Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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

share|cite|improve this answer

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 $25\text{cm}^2$

share|cite|improve this answer
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).

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.