Real water volume after sand, rock, and sump. Every dosing number on this site depends on getting this right.
Use 87.5 gallons, or 331 litres, for every dosing calculation on this site. A 75 gallon tank almost never holds 75 gallons of water. Rated size counts the rim, the sand, and the rock.
net = footprint 5,574 cm² × 53.3 cm ÷ 1000 − sand 13.2 L − rock 9.7 L + sump 56.8 L = 331.2 L
Volume is footprint times water height. One litre is 1000 cm³, one US gallon is 231 in³ or 3.785412 L. Everything else is the footprint formula for your shape.
The bowfront footprint is the back rectangle plus a circular segment. Given the chord (the tank length) and the bow height, the radius is R = (c²/4 + b²) / 2b and the segment area is R²(θ − sin θ)/2 with θ = 2·asin(c/2R). The often-quoted shortcut of two thirds chord times height is a parabolic approximation and runs about two percent high on a typical bow.
A sand bed is not solid. Aragonite has a grain density of 2.93 g/cm³, while a settled bed has a bulk density between 1.36 and 1.54 g/cm³ depending on grain size. CaribSea publish 85 lb per cubic foot for Special Grade and 96 for the sugar sized oolites. Bulk divided by grain density gives a solid fraction of 46 to 53 percent, and the rest is pore space still holding water. So a bed displaces about half its apparent volume.
Rock displacement follows the same logic and catches people out. Open pores flood, and water sitting in a flooded pore is already part of the system, so it displaces nothing. Only the mineral volume counts: a kilogram of dry reef rock at 2.8 kg per litre displaces 0.357 litres. Sealed voids that never flood add a little on top, which is why a chambered Pukani piece runs slightly higher.
That also means wet and dry weights are not interchangeable. Cured live rock weighed dripping carries roughly 30 percent pore water, so a kilogram of it holds only about 0.7 kg of mineral and displaces proportionally less.
The old hobby figure of ten pounds of rock per gallon of displacement works out to 0.83 litres per kilogram, more than double what the mineral density allows. It does not survive contact with the arithmetic.
If you want the number exactly right, weigh your rock, put it in a bucket, mark the water line, take the rock out, and measure how much water it takes to reach the mark again. That measurement beats any estimate.
Two other volumes people forget: the water sitting in the overflow box and the return plumbing, and the difference between the sump at running level and the sump when the pump stops. Neither is large, but both are real.
The honest way to check all of this is to fill an empty system with a measured container. It takes an afternoon and settles the question permanently.
Sources: Dieck, System Volume Calculator · CaribSea marine substrates · Webmineral, aragonite density · Aqueon tank dimensions (PDF)