Would you be able to post an original, raw image file?
For calculating the area … you can simply follow simple Segmentation techniques. Here is a helpful Segmentation workshop (and corresponding slides) that you might find helpful.
For the angle … you can select the measures you calculate in Set Measurements - using Fit Ellipse:
“Fits an ellipse to the selection. Uses the headings Major, Minor and Angle. Major and Minor are the primary and secondary axis of the best fitting ellipse. Angle is the angle between the primary axis and a line parallel to the X-axis of the image. The coordinates of the center of the ellipse are displayed as X and Y if Centroid is checked…”
and then use that angle relative to your vertical line. You might need to do a bit of scripting to get exactly the measure you want - so here are some links to get you started with that: