Algebra

Let the radius of the semi-circle  = r

The width of the window   = 2r

The length of one of the parallel sides =  

[12 ft - circumference of the semi-circle  - width  of window ]  / 2   =

[ 12  - pi*r - 2r] / 2

So .......the total area   =    area of the semi-circle +  area of the rectangular part of the window

We can express this as :

A  =  (1/2) pi*r^2  + 2r  [  12  - pi*r - 2r] / 2

A = (1/2) pi* r^2  + r [ 12 - pi*r  - 2r]

A =  (1/2)pi*r^2  +  12r  - pi*r^2  - 2r^2

A = 12r  - 2r^2 - (1/2)pi*r^2

A  = 12r  - [2 + (1/2)pi] r^2

If you have had Calculus, we can take the derivative of this, set it to 0 and solve for r to get to an answer

A'  =  12  - 2 [ 2 + (1/2)pi]r

A'  = 12  - [ 4 + pi] r

Set to 0

12 - [ 4 - pi] r     divide both sides by  [4 + pi ]

12/ [ 4 + pi]   = r   = about  1.68 ft

So...

The dimensions are

Width of the window  = 2r =  2(1.68)  = about 3.36 ft

Side of rectangle  = [ 12  - pi*(1.68) - 2(1.68)] / 2  = about  1.68 ft

Circumference of semi-circle  = pi (1.68) = about 5.28 ft