# Fixed area polygon maximize the area of its convex hull

For a fixed area, what kind of polygon that would obtain the maximum area of its convex hull.

For e.g. area 1 square, since it’s already convex, so the convex hull having area 1 too. Would there be any shaped polygon having area 1 but with its convex hull having area more than 1? If so what would be the shape you get the maximum convex hull area

You can construct a shape with limited area of any convex hull. Here's a simple solution. Points are $$(1,0),(s+1,0),(1,\frac{1}{2s}),\left(1,1\right),\left(\frac{1}{2s},1\right),\left(0,s+1\right),\left(0,1\right)$$
This is an octagon with area $$1$$ and convex hull $$(s+1)^2/2-\frac12$$
Another (easier) solution with a quadrilateral is $$\left(0,0\right),\left(a,0\right),\left(\frac{1}{a},\frac{1}{a}\right),\left(0,a\right)$$