A 10" container holds 2 1/2 gallons of growing medium. A 12" container holds 3 gallons of growing medium. Now, I'm curious about where you're located.
Every U.S. nursery I've ever dealt with sells the standard plastic nursery containers by both diameter measured across the top, as well as by volume. In fact, both measurements are molded onto the bottom of the plastic containers. Containers of 5 gallons and larger are usually sold by volume only, though.
But decorative containers are always sold by diameter measured across the top. Since you commented on the attractiveness of your container, I expected it would most likely be clay or ceramic and be marked in inches. Apparently, my bad.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams