Look at the usual equation: A = πr². Why is there no "2" there?
Let's derive it, and in particular, let's derive it from the onion proof, which is that a circle's area is composed of many small circles, arranged concentrically, like a 2D onion:
A = ∫_0^r 2πt dt
There's that blasted 2 again. The tau form is more beautiful:
A = ∫_0^r τt dt
Integrate it, and you'll get A = τr²/2, the constant being a result of the integral.
That is, to me, the usual equation is more properly A = 2πr²/2, the two 2s being different in their origins, and we just usually use & memorize the simplified form.
Another way to look at that would have been visible to non-calculus bearing ancients:
A = πr²
C = πD
… why do we arbitrarily use r in one equation, and D in the other? (… because we're using the wrong constant, and it bugs us, and we're sweeping that under the mathematical rug.)
Look at the usual equation: A = πr². Why is there no "2" there?
Let's derive it, and in particular, let's derive it from the onion proof, which is that a circle's area is composed of many small circles, arranged concentrically, like a 2D onion:
A = ∫_0^r 2πt dt
There's that blasted 2 again. The tau form is more beautiful:
A = ∫_0^r τt dt
Integrate it, and you'll get A = τr²/2, the constant being a result of the integral.
That is, to me, the usual equation is more properly A = 2πr²/2, the two 2s being different in their origins, and we just usually use & memorize the simplified form.