My, My! What a Pretty Valentine Pool We Have Here!

Geometry Level 5

After planning a cake, Mai Valentine is planning to build a heart-shaped swimming pool--like the one above--for Valentine's Day celebrations where a lot of her friends will attend. But she wants the pool to have some mathematical meaning and asks Joey Wheeler, one of the friends who is also an active Brilliant user, for help.

Joey gladly suggests an idea: add double caps like \frown\frown on top of a Reuleaux triangle:

  • The "double caps" are part of 2 small, identical, intersecting circles, which are the arcs LOOV\overparen{\small{LO}} \overparen{\small{OV}} in the diagram below.
  • The Reuleaux triangle (\big(the curved triangle ELV)ELV\big) is the intersection of 3 large circles centered at E,L,V,E, L, V, respectively.

More specifically, as shown above, the 2 small circles intersect at point OO lying on the circle centered at E,E, and are each tangent to one of the other two large circles at points LL and V,V, respectively.

If LV=1|LV| = 1 decameter (10 meters), what is the total area of the bolded Valentine swimming pool LOVELOVE in squared decameters (\big(in 100 m2)?\text{m}^2\big)?

Round your answer to the nearest 3 decimal places.


Before you try this problem, read this article about Reuleaux triangle.

This is the second chapter of the story. Check the following chapter directory if you are interested:

First - Second - Third - Fourth - Fifth

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