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!
emergent math 
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.