Brainteaser 1798: Trisection

From The Sunday Times, 2nd March 1997

We have taken a piece of paper in the shape of an equilateral triangle and folded it so that one of the corners lies at some point on the opposite side:

In the figure triangle A has an area of 36 cm²  and triangle B has an area of 16 cm².

What is the area of the unfolded piece of paper?

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant). The puzzle text above is taken from the book.