I could not find anyone else talking about this so I am convinced that my maths must be wrong but I seem to have found the equation: A = $\frac 12$rC. I came to this conclusion after discovering that the area of a circle divided by the circumference of a circle is equal to 1/2 of the radius. Rearranging that equation brought equations like the one above. I've looked around to see if anyone has used the formula but I the only equation for area using the circumference that I have found is A = $\frac{C^2}{4\pi}$. My first question is if my equation actually works; from my tests it works with all numbers up to 1000 excluding 1 and 2. If it does, is there a reason why the equation using pi is more commonly used than my equation above. I'm certain there is but I can not find it.
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1The area is $\pi r^2$; $C=2\pi r$ by definition of $\pi$. Then $\frac{1}{2}rC=\pi r^2=A$ so your formula will always work. Beyond that, I'm not sure what the question is. Your formula is fine - so is the other formula you presented. I suppose $C^2/4\pi$ is perhaps preferred because you don't need to know the radius, only the circumference (whereas for $\frac{1}{2}rC$ you need to know both radius and circumference) – FShrike Feb 18 '22 at 11:10
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This is the same as expressing the area in terms of the circumference. You're formula is correct and can be derived using the fact that $A = \pi r^2$ and $C = 2\pi r$. – soupless Feb 18 '22 at 11:10
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You know, Jake, the whole idea of algebra is that one quick algebraic computation can take the place of $1,000$ numerical computations. – Gerry Myerson Feb 18 '22 at 11:20