I can't seem to come up with a formula for the area of this triangle. This should be a calculus problem, but any formula that I can come up with does not require calculus to get the answer. The best formula that I have come up with so far is

$\displaystyle

(1/2)*sqrt((1/2+x)^2+(1/2-x)^2)*(1/2)

$

Problem:

Consider a square piece of paper with sides equals to 1 unit. We label the four vertices as ABCD. Now inscribe a quarter circle with radius of 1 unit such that its center is at vertex A. (See figure 1). Next, we fold corner labeled as vertex A to touch the circumference of the quarter circle. We want the fold to create a triangle (the folded part forms a triangle; see figure 2). Note that the end points of the crease have to be on side AB and side AD in order to do this. Our task is to find the exact area of the smallest and the largest of these triangles created in the above process.