What’s the average length of chords at a given angle in a rectangle? Articles are published on average length of chords on circles, squares, rectangles etc..like [1] where they considered either random chords or chords at all angles in a regular geometry, but I could not find a result for average length of chords which are at a prescribed angle in a rectangle, refer the figure for a clear idea. How would you derive an expression for that?

[1] Kuchel, P. W., & Vaughan, R. J. (1981). Average lengths of chords in a square. Mathematics Magazine, 54(5), 261-269.

For any shape, the area is obtained as the height times the average horizonal chord length.
Hence for a rotated rectangle, the average horizontal chord length is 
$$\frac{WH}{|H\cos\theta|+|W\sin\theta|}.$$
Note that the average value on $\theta$ is $$\frac{2WH}\pi\left|\text{artanh}\left(\frac HD\right)-\text{artanh}\left(\frac WD\right)\right|$$ where $D$ is the diagonal and this is the average chord length regardless the orientation.



For a circle,
$$\frac{\pi r^2}{2r}.$$
For an ellipse,
$$\frac{2\pi ab}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}}.$$
For an arbitrary triangle,
$$\frac{\begin{vmatrix}x_1-x_0&y_2-y_0 \\ x_2-x_0&y_2-y_1\end{vmatrix}}{2(\max(y_0,y_1,y_2)-\min(y_0,y_1,y_2))}.$$
If you rotate it, the area remains the same and the ordinates are replaced as
$$y_k\leftrightarrow y_k\cos\theta+x_k\sin\theta.$$
