Ha. Funnily enough my father tacked this on to an email I received from him this morning, coming on the heels of my defense of the real-world.

I enjoyed this decidedly non-real-world problem and thought you might too.

Consider a unit square that encloses a unit equilateral triangle:

As shown, the area of the square is 1, and the area of the triangle is sqrt(3)/2. What is the largest equilateral triangle that can be inscribed in the square?

Happy problem solving, everyone!

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Symmetry got me one vertex at a vertex of the square.

the rest is based on cos(30) = 2 x square(cos(15)) – 1

no tables !

result : side of triangle is 1.0718 or so

area = (root 3 over 4) x side squared

Hello again.

How about the largest square inside an equilateral triangle ?

Might need to solve an equation for this one.