# Surface area of a sphere alternative

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.

• See the tau manifesto versus the pi manifesto for similar arguments/concerns. The end result is that both are correct, and whichever you use is your own personal preference. – JMoravitz Apr 24 '17 at 20:14
• Honestly using diameter over radius only has practical applications as it's easier to measure the diameter in practice (hence often formulas in engineering refer to diameters of pipes/cylindrical objects) – Triatticus Apr 24 '17 at 20:25

## 2 Answers

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.

• That's kind of annoying, but at least I'm not wrong ;) Thanks! – Bobdabiulder Apr 24 '17 at 19:54
• From a purely aesthetic point of view, generally using $r$ seems nicer since it avoids fractions where possible, for example in $\pi r^2$ as opposed to $\frac{1}{4} \pi d^2$ – LtSten Apr 24 '17 at 20:00

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!

• Why can't you go from circle to sphere? What do you mean? Also, in what case does a sphere exist in >3 dimensions? Experimental physics? – Bobdabiulder Apr 24 '17 at 19:57
• Dimensions greater than 3 is just a natural extension to euclidean space, and as such it's natural to ask what the surface area of a hypersphere is. Sometimes optimization problems involve higher dimensional spaces because of a large amount of conditions/variables – Triatticus Apr 24 '17 at 20:23