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Isoperimetric Quotients
By Ystrad Farm on February 23, 1998:

I am doing my GCSE coursework and I have to find out about isoperimetric quotients. I have the formula to work it out for plane shapes and have found out that the I.Q refers to shapes of equal perimeter. I just need more information about its uses.Please help!!

Abi

By Sujata Krishna on March 18, 1998:

Hello Abi,

Sorry that your question has been unanswered for so long. In the spirit of better late than never, here are some thoughts that you might want to think about:

  • Among all planar shapes with the same perimeter the circle has the largest area.
  • Among all planar shapes with the same area the circle has the shortest perimeter.
  • Among all triangles with the same perimeter, the equilateral one has the largest area.
  • Among all quadrilaterals with the same perimeter, the square has the largest area.
  • In particular, among all rectangles with the same perimeter, the square has the largest area.
  • Among all planar curves of fixed length with fixed endpoints, a circular arc encloses a maximum area between it and the line joining its endpoints.

Hope you find these points interesting to ponder over,
Sujata.