There is no absolute magic number that will guarantee that you will never have a problem. There are a lot of different factors involved. Coop size, climate, and the personality of your individual chickens are important ones, but there are others. The rule of thumb on this site is that if you have a coop that gives 4 square feet per chicken, you need 10 square feet per chicken in the run. Most building materials come in 8' or 4' lengths, so those are usually good dimensions to start with. You can usually build something with those increments with minimum cutting and cost. Your building techniques and what you have to work with also enter in. If you are setting posts and stretching wire in between, 8' is usually a good distance to space the posts.
For ten chickens, the minimum I'd go with is 8' x 12'. That is slightly less than the rule of thumb but close enough. I strongly agree that more is better, but you would probably be OK with this size if your coop is of decent size. If you build the minimum, you have no room to increase your flock later without more building. I suggest you consider that.
sounds like you planning on covering your run? If so, I suggest looking at the standard dimensions of your roofing material and plan for a little overhang. That will help keep your run drier.
What is the width of your coop? An easy way to build a run is often to make it as wide as your coop and let your coop be one end or side of the run. So for 10 chickens, you need a minimum of 100 square feet. If you divide the width of your coop into 100, that gives you the minimum length you need for your run.
Say your coop is 6 feet wide. You divide 100 by 6 and you get 16 and a fraction. So a run 6' x 16' would work, although bigger is always better.
Say your coop is 4 feet wide. 100 divided by 4 is 25 feet, but that is usually not very practical. I like to have a run wide enough so a chicken can pass a bully and get to the food, water, and nest box without challenging the bully. For me, with 10 chickens, 6 feet is the minimum width I'd go.
Hope this helps. Good luck!