Why are people told the formula for the circumference of a circle

>>8813397
The radius is a more fundamental measure.

>>8813397
Because then you can integrate the circunference to get the area.

>>8813397

...And particularly where the latter is the literal definition of π.

Real talk: I think it has to do with presenting extremely simple information at the late primary/early secondary educational level, OP. When you only speak of the radius, Then you can immediately phrase everything in terms of it without bogging the student down in conversion factors of 2. Of course, you could just as easily accomplish same by consistently sticking with the diameter, but I think that /sci/ will agree that the radius is a more "natural" consideration of circles and spheres.

First, express the following, respectively, in terms of the radius of each: the circumference (or, perimeter) of a circle, the area of a circle, the surface area of a sphere, and finally the volume of a sphere. We repeat the familiar formulae

[math] C = 2 \pi r \;\; ; \;\; A = \pi r^2 \;\; ; \;\; A = 4 \pi r^2 \;\; ; \;\; V = \frac{4}{3} \pi r^3 [/math]

Now do it again, replacing radii with diameters, and we have

[math] C = \pi d \;\; ; \;\; A = \frac{1}{4} \pi d^2 \;\; ; \;\; A = \pi d^2 \;\; ; \;\; V = \frac{1}{6} \pi d^3 [/math]

In terms of complexity/simplicity, this is almost a total wash, and so it's almost down to choice at this point, if one wants to consistently stick with one measurement or the others for the purposes of simple instruction. Notice how in the move from the radius rules to the diameter rules, although the circle-circumference and the sphere-surface-area rules do simplify slightly, the circle-area rule adds back (slight) complexity in the form of a fractional term. Meanwhile, in the case of the radii, the only fractional consideration which has to take place is with the sphere-volume rule. Everything else is presented as it were in "numerator" style, top-side. The student then only has 2-4 concepts that he has to master, to use the rules: simple algebra, what pi is (just some number), what the radius is, and plugging in values.

>>8813419

I understand exactly what you're trying to say, but I think that you can phrase it better. There is a clear sense in which neither radius nor diameter is either one more "fundamental" than the other; it is ridiculous, for example, to speak of either object /preceding/ the other. Both exist simultaneously with respect to any circle, or sphere.

I instead repeat my phrase that the radius is a more "natural consideration" of these shapes, among people, in practical considerations. Obviously the diameter will come up, but not quite as much.

>>8813397
Because 2 is multiplying pi, you dummy. That's why we invented Tau.

>>8813441
Fuck off Michael Hartl

>>8813397
More importantly, why are they told 2*pi*r instead of tau*r?

>>8813460
Because you're ultimately just pushing the scalar you don't like somewhere else.

If you insist on working with diameters then whenever you try to do analysis with a ball of diameter d you have to divide everything by 2.

I'd rather have a factor of 2 in my simple geometry than a factor of 1/2 when I'm trying to juggle multiple epsilon-delta limits

>>8813441
>>8813460

t. Hartl

This thread smells like disgusting nominalist post-18th century interpretations of geometry.

>>8813439
>it is ridiculous, for example, to speak of either object preceding the other

Don't be a pretentious twat. The radius always precedes the diameter.

>>8813397

OP,

Ignore all the other answers, they have no idea what they're talking about.

You construct circles with compasses. The distance between the pin of the compass and the pencil is the radius of the circle which you are about to draw. There isn't a simpler way to construct circles, which is why the radius is more fundamental than the diameter. You construct a diameter from the radius, not the other way around. Any tool that has a midpoint explicitly relies on a simpler tool with a radius to construct that midpoint, so diameters can't be more fundamental than radii.

Also this belongs in /sqt/
>>>8812892

>>8813546

What you present as a philosophically satisfactory distinction actually isn't one, though. And the reason why this is so is because it is always possible to speak of circles in terms of their diameters as opposed to their radii. /Circles always have both items associated with them, at all times/. You are making the mistake of thinking that just because it is more natural for humans to think of the one than the other, that this then means that the one really is more "fundamental" in some deep sense.

I can just as easily propose: "take a length and spin the length about its midpoint; the figure described by the length's endpoints is a circle". At which you can readily miss the point by objecting that the midpoint implies the radius, the radius is smaller than the diameter perhaps, or similar statements. But these miss the point: a circle has a radius and a diameter /all at once/, and each are used at various times.

It is probably best to say that the radius and the diameter are /dual/.

>>8813546

Moreover, I see a nice piece of pedantry which literally supports my alternative model! The spun diameter describes the circle in one /half-turn/, while the spun radius requires twice as much of a turn, or one full turn, to do the same job. Here we have a clear demonstration of a sense in which a circle may be described more simply by means of a diameter, instead of using a radius.

At THIS point, I should expect that you would further object about the endpoints of the two half-arcs adding complexity, and you'd have something. But I would refer you back to >>8813433 where I indicate that either way, there is roughly a wash in terms of complexity.

>>8813397
I was taught pi * d... Did I go to a brainlet school?

>>8813397
Because it's actually [math] \tau r [/math], not [math] 2 \pi r [/math] faggot.

>>8815280
and the surface is [math] \frac{1}{2} \tau r^2 [/math]

This makes a lot more sense

>>8815282
No. The surface area is [math]\frac{1}{8}\tau d^2[/math]

I was always taught it as C=pi*D

>>8815280
>>8815282
>>8815457
>irrationals

JUST