Wolfram Alpha mentions the following general formula for the area and perimeter of hypocycloids given the number of cusps, n.
My issue is I've seen Wikipedia that references this page but mentions a slightly different formula.
I've checked other references and they mention the same formulas that in Wikipedia. The derivation from the general formula was clearly shown in Wolfram Alpha. So why is there a contradiction?
I've tried applying the same formula for an astroid, where there are 3 cusps. I manage to get the answer.
I don't get why in deltoids the general formula does not work.
https://en.wikipedia.org/wiki/Deltoid_curve#cite_note-Weisstein-2
There's no contradiction, just a difference in notation. In the Wikipedia article, $a$ is used as the radius of the rolling circle, while in the Wolfram MathWorld article, $a$ is used as the radius of the outer circle. Using Wikipedia's convention, we have $3a$ as the radius of the outer circle, so the formula given on MathWorld for the arc length of the deltoid yields
\begin{equation*} s_{3} = \frac{8(3a)(3-1)}{3} = 16a, \end{equation*}
consistent with the Wikipedia article. For the area of the deltoid, the formula given on MathWorld yields
\begin{equation*} A_{3} = \frac{(3-1)(3-2)}{3^{2}}\pi(3a)^{2} = 2\pi a^{2}, \end{equation*}
again consistent with the Wikipedia article.