House shapes

These two houses have equal volume. One of them is a hemisphere, and one of them is a cube. Which one is cheaper to heat during winter? The answer is the hemisphere. The reason it is cheaper to heat is because it has less surface area and because heat loss increases with more surface area. Using high school geometry, you can compute what radius the hemisphere that gives volume V. You can then compute the surface area of a hemisphere of that radius. You can also compute what length of cube gives volume V. You can then compute the surface area of the 5 exposed sides of that cube. You'll see that the hemisphere has about 25 percent less surface area than the cube of equal volume. This is an example of the isoperimetric inequality.