Thanks to Ahmad from Jordan and Atiksh from India for the suggestion!

In the diagram below, the points P, Q, and R are centers of arcs passing through O, the center of the circle. If the arcs PQ, QR, and RP are equal, and PO = 6, what is the area of the shaded region?

As usual, watch the video for a solution.

Area of a flower

Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Area Of A Flower

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

The flower has 3 petals. Focus on the top petal. Let S be the top of the petal between P and Q.

Since P, Q, and O are centers of circles passing through each other’s centers, the circles all have the same radius. Since PO = 6, we have 6 = PS = OS = QS = QO. Then triangle PSO is an equilateral triangle, and the sector PSO has a central angle equal to 60 degrees.

We can then calculate half the area of a single petal as follows:

half petal = circular sector – equilateral triangle

The area of half of a petal is:

62π(60/360) – (62√3)/4
= 6π – 18√3

The area of a single petal is double this, and we need the total area of 3 petals. So we multiply this area by 6 to get the total area:

total area
= 6(half petal)
= 6(6π – 18√3)
= 36π – 54√3
≈ 19.57

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PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

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