38715

An Appearing Act:
I believe the mystery of where the increase in area comes from the inaccuracies of cutting paper with simple scissors and careful positioning of the paper shapes. The area of the rectangle does not increase when the shapes are rearranged, it just looks like the area has increased. The shapes don't actually line up properly when positioned together in the format in the image, there are spaces and it is when you include the spaces in the measurement for the area you find that the area seems to have increased (This is easier seen when you draw out the triangles in graph paper in the arrangement depicted in the question's photo).
Moreover, it is clear once you start looking at the gradients of the edges of the shapes after cutting. If the pieces of paper were to fit together the gradients of the edges of the papers touching together would have to be the same. However, if you measure the gradient of the 2 larger shapes at the top, it has a gradient of 5/2 (which becomes -2/5 when rotated 90 degrees which it is in the new positionings (parallel lines must multiply to -1)). The the gradient of the hypotenuses of the lower triangles is -3/8 (the triangles aren't rotated in the new positionings, so we don't have to make any changes to that). Now if you compare the gradients, we have -2/5 and -3/8. Now these are not the same. If those shapes were to fit perfectly together they would have to be equal. The trick in this problem is that -2/5 and -3/8 are actually very similar gradients (this is easier seen in decimal, -0.4 and -0.375). Similar enough to look like they are the same when positioned roughly together, so you don't see the gaps.
In conclusion, the reason this works is because the gaps between the shapes are small enough you don't notice them. Furthermore, this trick can easily be repeated, you just need to make sure that the shapes you cut up have a similar enough gradients that the viewer won't spot the gaps.