I've heard there's a way to measure a flat foot in degrees. How do I do this?

You may be referring to the navicular drop test (NDT). In this test the person stands with the feet about shoulder width apart. The navicular tuberosity is felt and marked. This is a boney projection along the side of the navicular bone. The navicular bone is the bone on top of your foot above your arch.

With the ankle joint in the middle (neutral), the distance from the navicular bone to the floor is measured. The person with a flat foot then relaxes the foot and the same measurement is taken again. The difference between these two measures is called the navicular drop.

Movement of 10 mm or less is considered "normal." More than 10 mm suggests too much rolling in of the ankle and drop of the arch called pronation. Pronation isn't really divided into degrees except perhaps mild, moderate, or severe.