# Area of a quatrefoil inside a circle

What is the maximum area of a quatrefoil that is inscribed in a circle of radius 6?

My first guess was to cut the circle into small regions but that doesn't seem to work. The solution does not have to be simple.

• According to Wiktionary, a quatrefoil is a "symmetrical shape that forms the overall outline of four partially-overlapping circles of the same diameter". However, the degree of overlap may vary, so unless you are more specific, your question cannot be answered. Mar 5, 2019 at 2:43
• @FredH It is quite possible that he wanted to mean the overlapping degree is $90^\circ$ because only four symmetrical shape of partially overlapping circles are found in Google. I don't see further any. And more specifically, A quatrefoil is an ornamental design of four lobes. Mar 5, 2019 at 2:50
• Thank you for your correction, I included a picture that will probably help. Mar 5, 2019 at 2:56
• @BorKari It would be helpful if you could give a precise geometric or coordinate description of this shape. Mar 5, 2019 at 3:02
• @BorKari You mean to say the maximum area of the quatrefoil? Mar 5, 2019 at 3:02

Here, we see the symmetrical shape of four partially overlapping circles at the point $$A, B, C$$ and $$D$$ respectively. Connect those points and you will get a square having side equal to the diameter of small circle and that is obviously equal to 6. Let denote the area of the square $$ABCD = [ABCD]$$.
So, the area of the quatrefoil = $$[ABCD]$$ + The area of 4 semi circles = $$6^2 + 4*\frac{1}{2}*\pi3^2$$ = $$36 + 18\pi \approx 92.549$$
If I understand the intended shape correctly, its outline is formed by four semicircles of radius $$3$$ whose centers form a square with a diagonal of $$6$$ units. They intersect at their endpoints, which form a larger square with a side of $$6$$ units. The total area inside the quatrefoil is then the area of the square plus the area inside the four semicircles: $$6^2 + 4(\frac12\pi\cdot 3^2) = 36 + 18\pi$$.