How to calculate how much a dump truck box can carry?

if you want to know the approximate volume... multiply the length by the width, and then the result by the height.

If you want to know the weight... well, multiply the volume by the density of whatever it is you're trying to carry, and then you'll know how much that volume of "stuff" weighs.

Note that just because the box will hold that much doesn't mean the truck can actually carry that much. The truck will have a weight limit which, for dense materials, can be considerably less than will physically fit in the box. If you rent a moving truck or a trailer, somewhere on it there will be a load limit. It's surprisingly easy to exceed this... normally, a couple of inches of sand in the bottom will do it.

If you're simply looking for how much weight the truck can legally carry, you simply subtract the allowable gross weight (for a tandem axle, it's typically 54,000 lbs), minus the tare (empty) weight. For most tandem axle dump trucks, you can expect 15 to 16 tons of allowable net weight. For tri-axles, quad axles, quints, and centipedes, state laws vary far and wide with what they'll allow for each additional axle.