Another method of finding area of hypocycloids

Question

I was finding the are the of hypocycloids. Then it struck me that apart from integration, there could be another method of finding the area of the hypocycloid with different curves. But the problem is I am not getting my answer right. So could somebody please help me tell if my logic is wrong altogether or if I am doing some other mistake.

Here is the another method I am talking about:-

Take case of a deltoid - we can make a deltoid by taking an equilateral triangle, and on all three of its vertices drawing circles whose radius is half the side of the triangle. Now the figure left in the middle is a deltoid. Similarly, we could use n number of sides in the regular polygon to draw hypocycloids with n cusps. (see the pictures below - the red coloured drawing in the middle is the hypocycloid)

Is this idea wrong? Thank you so much :)

Answer 1

I think you assume that hypocycloids consist of circle arcs, but this is not the case:

Consider e.g. the Astroid:

It is an algebraic curve of degree 6, defined by $$ (x^2+y^2-a^2)^3+27a^2x^2y^2=0 $$ which is nowhere close to circle.

To come back to your example: The Deltoid is defined by this equation of degree 4:

$$(x^2+y^2)^2+18a^2(x^2+y^2)-27a^4 = 8a(x^3-3xy^2)$$ which also is certainly not composed of circles.

Answer 2

I used a software to produce the hypocycloid and then I tried this with an Astroid. Here is what happened:

Hypocycloid been drawn

As you can see, the circumference of the circle doesn't go over the shape of the hypocycloid. It doesn't work like that unfortunately.