The 10.50 inches is the tire's advertised width, usually taken at the tread, not the sidewall bulge. And the 15 is the size of the wheel it's designed for. So, in this case you've got a 31-inch ...

Using 235/55R18 as an example, 235 is the tread width in millimeters. The higher that first number, the wider the tire. The 55 is the height of the tire expressed as a percentage of the width we ...

Finally, there is no direct correlation between section width and tread width (the actual tire ground contact patch). Tires with identical section widths may have slightly different tread widths.