Smallest convex polyhedron containing integer points of a cylinder

Question

A cylinder has height $6$ and radius $3$. The centers of the two bases are $(0,0,0)$ and $(0,0,6)$. Find the volume of the smallest convex polyhedron that encloses every lattice point inside the cylinder.

Answer 1

Let's look at the cross-section of the cylinder on the $x$-$y$ plane:

Now it's obvious that I just drew a picture here, but in concept it's easy (if annoying) to verify which lattice points lie within the circle.

The green region represents the convex hull of the points. To find the area of this region, it's easy enough to do some area addition arguments, but I'm going to take the opportunity to apply Pick's theorem: $$A=i+\frac{b}{2}-1=21+4-1=24$$

Now, the convex hull of all the lattice points in the cylinder is just going to be a prism with this base (by symmetry) and a height of $6$. This gives a total volume of $144$.

Answer 2

Draw the base in the xy plane. Draw a convex polygon around the lattice points within the circle. Calculate the area of the polygon by splitting it up in squares and triangles. Multiply the area by the height of the cylinder.