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Forgive me if this question is a little too strange or maybe even off. Mathematics has never been my strong point, but I definitely think it's the coolest...

Anyway, I was looking into tau, pi's up-and-coming sibling. I started rethinking why pi worked. The thing about tau is that it supposedly skips the step of doubling πr, since tau is twice pi. I tried this, and it works! (Of course it works. The fact that this surprises and amazes me shows how little I get out...)

Then I started wondering why we do 2πr instead of πd. It does give the same answer... I checked. Is there any reason why using the radius is preferred over using the diameter?

Here's my work:

r = 6 (ergo diameter must equal 12)

C = 2πr

C = 37.68

and...

C = Tr

C = 37.68

and...

C = πd

C = 37.68

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    $\begingroup$ Only for historical reasons. If you know $\pi$ then you know $\tau$. If you know radius, then you know diameter. Read more at this related question $\endgroup$ – JMoravitz Jul 21 '16 at 1:54
  • $\begingroup$ The area of a circle is $A=\pi r^2=\frac{\tau}{2} r^2$ you don't get to skip squaring the radius. $\endgroup$ – n1000 Jul 21 '16 at 1:55
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    $\begingroup$ @GeneralNuisance There are a lot of formulas that are easier to work with in terms of $r$. Even in calculus there is a differential $dr$ when talking about polar coordinates. That is the same $r$ as when talking about concentric circles with the Origin as center and some point $P$ on it. Surface area formulas of revolutions (about x or y-axis) depend on a (variable) $r$ and not on some $d$. And so there are many reasons $\endgroup$ – imranfat Jul 21 '16 at 2:32
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    $\begingroup$ @imranfat I think this could be incorporated into a very good answer. $\endgroup$ – General Nuisance Jul 21 '16 at 2:45
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    $\begingroup$ "skips the step of squaring $\pi r$, since $\tau$ is twice $\pi$": Do you mean doubling instead of squaring? I think this may be why n1000 mentioned the area formula, since this involves squaring (regardless of whether $\pi$ or $\tau$ or $r$ or $d$ is used). $\endgroup$ – stewbasic Jul 21 '16 at 2:57
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You can define a circle knowing the centre and the radius (distance $r$).   A circle is the set of all points, on a 2D-plane, at distance $r$ from the centre.

That's a concise and elegant definition; try doing so using the diameter (distance $d$).

Then, having defined circles using the radius, it becomes convenient to also define the radian measure of angles in terms of the radius of a circle.   One radian is the measure of an angle subtended at the centre of a circle by an arc length equal to the radius.

Then we asked: what is the radian measure of a straight angle (that formed by two rays of a line)?   Why it is that irrational number we have decided to call $\pi$, to honour Pythagorus.

What then is the angle subtended by the circumference of a circle?   Well, we could call it $\tau$, but it is $2\pi$, and we just happened to have named $\pi$ first.

Thus $$\begin{array}{cc}C&=&2\pi r &=& \pi d &=& \tfrac 12 \tau d &=& \tau r & \text{circumference of circle}\\[1ex]A&=& \pi r^2 &=& \tfrac 14 \pi d^2 &=& \tfrac 18 \tau d^2 &=& \tfrac 12\tau r^2 & \text{area of circle/disc} \\[2ex] S &=& 4\pi r^2 &=& \pi d^2 &=&\tfrac 12 \tau d^2 &=&2\tau r^2 & \text{surface area of sphere}\\[1ex] V &=& \tfrac 43 \pi r^3 &=& \tfrac 16 \pi d^3 &=&\tfrac 1{12}\tau d^3 &=& \tfrac 23 \tau r^3 & \text{volume of sphere/ball}\end{array}$$

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