3 ms·

Area of a circle?

• deathanatos 2y ago
  Yes, even that one gets more beautiful, too. 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.
  • hinkley 2y ago
    Unfortunately the ancients didn't invent calculus. Pi had been in use a long time when Liebniz and Newton came along.
    • deathanatos 2y ago
      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.)