Grand Canyon: 230 Miles by road from North Rim to South Rim
And a topographic map of the Grand Canyon, which shows clearer all the different side canyons:
Topographic Map of the Grand Canyon
From our own experience, and what we have since heard from some fellow travellers: don't be discouraged from trying to get a campsite at the only campground of the North Rim - you might be lucky even if it is supposed to be booked out! For our night this was the case, and eventually there were almost 15 vacant sites all night (one right next to us).
You can also camp in the Kaibab National Forest just outside the park's entrance; in this park "free camping" is permitted, meaning, you can camp in the wild providing you stay a ¼ mile from any water (lake, river, etc.) and don't start a forest fire or leave rubbish behind...