Define the width of the window as W, the height as H, and the area as A.
A = HW + π (W/2)2/2.
However, the perimeter is 28, so 28 = 2H + W + π W/2
Solve the second equation for H and substitute that into the first equation. This gives you an expression for the area in terms of the width of the Norman window.
Next, take the derivative. Then set the derivative equal to zero and solve for W. Once you know the width, substitute for W in the second equation above to get the height.
Hope this helps,
Stephen La Rocque.