Using 215/60R15 as an example,..... the "215" is tire width in millimeters(25.4mm = 1") measured at the widest bulge of the sidewall(known as "section width") when mounted on its recommended rim width(7" in this case). 215mm is 8.46". The "60" part of the number is sidewall height in percentage(known as "aspect ratio"), a 60 series tire is 60% as tall as it is wide, in this case 5.08"(60% of the 8.46") from bead seat to top of tread. Overall tire height is the 15" rim plus TWO sidewall heights(one top, one bottom)5.08" x 2 =10.16", for a total height of 25.16".
This is how the math is, but tires true sizes vary a good bit from brand to brand. Don't ask me why, it is a pet peeve of mine, the tire companies have a standard for measurement but tires seem to vary (up to an 1" in both height and/or width) for the same size between two different companies.
The 215/60R15 in this example will be slightly wider at the sidewall when on a 8" rim, maybe 8.75" instead of 8.46". If you look tires up in a catalog they all give a recommended(or measuring rim) width for each size tire.
In your case it is not the width that is the total problem, it is the height, which is why I suggested a low profile percentage like 50 or 55 to get it to clear. A 215/50R15 will still be 8.46" wide across the sidewalls on a 7" rim but will only be 23.46" tall because of the 50 series profile.