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I understand that a circle will have the largest area for a given perimeter but I don't get The smallest area. Is it a triangle because it has the least amount of sides so must be smaller?

Plasman
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    Depends on how you define shape. If lines are allowed, then they have the minimum area of $0$. Otherwise no shape has the minimum area - you can stretch a "shape" to make its area as small as you like. – Ningxin Jun 24 '20 at 12:20
  • @Plasman, did you see any pattern? Like increasing or decreasing? (Though it's obvious that it should be increasing as number of sides increase) – UmbQbify Jun 24 '20 at 14:25
  • I mean if they have a regular shape that is complete. I experimented with a regular: triangle, square, pentagon, hexagon, and a decagon. They all had a perimeter of 60cm. For the triangle, the area was 173.205081 cm2. With the square, it was 225cm2. With the pentagon, it was 247.7487cm2. With the hexagon, it was 259.808cm2, With the decagon, it was 276.9915cm2. So that's why I think that triangle has the smallest area. – Plasman Jun 24 '20 at 14:34

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There is no lower-bound to the area of a shape with given perimeter $C$. To prove this, we can take a rectangle with sides $\epsilon$ and $(C/2-\epsilon)$, and area $\epsilon(C/2-\epsilon)$. As $\epsilon \rightarrow 0 $, the area goes to zero.

Rd Basha
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