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.
Thank you for your time.
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asked Mar 5, 2019 at 2:24
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2$\begingroup$ 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. $\endgroup$– FredHMar 5, 2019 at 2:43
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$\begingroup$ @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. $\endgroup$– Anirban NiloyMar 5, 2019 at 2:50
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$\begingroup$ Thank you for your correction, I included a picture that will probably help. $\endgroup$– Sina Babaei ZadehMar 5, 2019 at 2:56
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$\begingroup$ @BorKari It would be helpful if you could give a precise geometric or coordinate description of this shape. $\endgroup$– Isaac BrowneMar 5, 2019 at 3:02
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$\begingroup$ @BorKari You mean to say the maximum area of the quatrefoil? $\endgroup$– Anirban NiloyMar 5, 2019 at 3:02
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2 Answers
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If it's a quatrefoil, then four circles have the same diameter and to be inscribed in the large circle, their diameter have to be equal to the radius of the large circle.
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 = $6^2 + 4*\frac{1}{2}*\pi3^2$ = $36 + 18\pi \approx 92.549$
And that's your answer.
answered Mar 5, 2019 at 3:21
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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: $6^2 + 4(\frac12\pi\cdot 3^2) = 36 + 18\pi$.
