How to solve this optimization problem with the sun

Question

I want to build a square awning that shades my kitchen window. What are the optimal dimensions for this awning and the angle it makes with the rectangular kitchen window (which is embedded in a wall perpendicular to the ground)?
My first thoughts:
Let $\vec S$ be a point indicating the position of the sun (restricted to half of the northern hemisphere) and $\vec a$ a point on the awning. Also, $\vec p_0$ is a point on the window and $\hat n $ is a vector perpendicular to the window. Then $$((\vec a-\vec S)t+\vec a-\vec p_0)\cdot \hat n=0$$ can be solved for $t$ to obtain the coordinates of the point's shadow on the window: $$w(\vec S,\vec a)=(\vec a -\vec S)\frac {(\vec p_0 -\vec a)\cdot \hat n}{(\vec a -\vec S)\cdot \hat n}-\vec a$$
We can therefore define $W(\vec S)=\{w(\vec S,\vec a) : \vec a \text {is in the perimeter of the awning}\}$, it seems natural to venture that $W$ forms a closed curve in 3D space confined to the window plane. Denote the interior of this closed curve by $I$ and let $P$ be the set of points living in the window plane that are also part of the window.
Is there a way to find the lengths of the sides of the awning, as well as the angle it makes with the wall such that $P\subseteq I$ and $\text{Area}(I)-\text{Area}(P)$ is minimized?
I was thinking that maybe the calculus of variations could yield an expression for these quantities but to be honest, I have no experience with the topic. It could be possible to restrict the position of $\vec S$ to a smaller subset of the sphere (I am after all on Earth). If there are simpler ways to obtain a solution, they are more than welcome.