[jay_shark]

Here is why DA=wdh+hdw is wrong

Label the rectangle A with vertices (0,0) (w,0) ,(0,h) (w,h)

Now Label the larger rectangle which vertices (0,0) (w+dw,0) (0,h+dh) (w+dw,h+dh)

Label the smaller rectangle with vertices (0,0) (w-dw,0) (0,h-dh) (w-dw), (h-dw)

Now we use an incorrect fact that
2DA = the larger rect - the smaller rectangle
2DA= (w+dw)(h+dh)- (w-dw)(h-dh)
2DA= wh +wdh +hdw +dwdh -[wh-wdh -hdw +dwdh]
2DA = 2wdh +2hdw
divide both sides by 2
DA= wdh +hdw