I am no math major, and maybe that explains it, but when figuring the area of the triangle I come up with 32.5, but when adding up the area of the individual shapes I come up with 32. How can this be?

Neither large shape is a triangle, first of all, because those two smaller triangles aren’t similar (The little green triangle has two shorter sides of 2 and 5, therefore the sine of the sharpest angle is 2/5. The bigger red triangle has two short sides of 3 and 8, therefore its sharpest angle has a sine of 3/8.).

In the top shape, the longest ‘side’ bends slightly down; in the lower one, it bends slightly up.

The difference between the two is one square worth of area.

Look at the point on the boundary of the bottom figure where the dark green and red triangles touch. The corresponding point in the upper figure is clearly outside the figure. Likewise, where they touch in the upper figure, the corresponding point in the lower figure is clearly inside the figure.