Surface area of a sphere alternative

Question

Every source I find lists the surface area of a sphere as $4\pi \times r^2$. Couldn't you write it as $D^2\times \pi$? This works every time I tested it, and it's a simplified version of the original equation. Why do mathematicians make it so much more complicated? I remembered proving the $4\pi \times r^2$ back in math class years ago, and I asked the teacher why wouldn't the diameter squared times pi equation work. She couldn't find a reason why it wouldn't! Please explain.

Answer 1

As the diameter is twice the radius ($d=2r$), obviously: $$ \pi\cdot d^2=\pi\cdot(2r)^2=\pi\cdot 4\cdot r^2=4\pi r^2 $$

But mathematicians generally consider the radius more fundamental than the diameter, probably to a non-insignifant degree because of tradition, but as an example the radius occurs naturally in the equation of a circle, and therefore we prefer to express formulas using the radius.

Answer 2

A sphere is more cleanly discribe as the set of points which are at a distance $d \leq r$ from some fixed "center" point, where $r$ is the radius of the sphere.

You can of course describe a sphere in terms of its diameter $D$, but unless you cheat and use the radius (as in "the set of points witin $D/2$ of some fixed point") the description becomes much more difficult.

"A sphere is a set of points such that the distances between each pair of points is no more than $D$." It is true that this definition is equivalent to the usual definition of a sphere, but that is not a trivial fact!

Since the radius is a more natural property of the sphere, formuas for the volume and the surface area are usually expressed in terms of $R$, not $D$.

However, it is perfectly fine to say that $A = D^2 \pi$. Just don't think you can go from circle (circumference$ = D\pi$) to sphere (surface $= D^2 \pi$) to hypersphere (hyper-surface $= D^3 \pi$) because that last statement is simply not true!