How to Figure the Volume of a Round or Rectangular Tank
Why this matters
A tank's volume sets the chemical dose for a shock treatment, the size of the pump-out, the run time of a transfer, and the answer to "how much is left" on a fuel or water tank. The nameplate capacity is often missing, painted over or describes a tank that was replaced. So the tech measures it and does the arithmetic on site, and the two errors that cost the most are both quiet ones: an outside measurement used where the inside belongs, and a straight-line guess at how much is in a horizontal tank that is partly full. This card gives the two formulas, the order to measure in, and why the horizontal partial fill is the one case where you stop calculating and read the maker's chart.
The two formulas, with the constants traced
Vertical round tank: gallons = 5.875 x D x D x H, with the inside diameter D and the water height H both in feet.
Where 5.875 comes from: the area of a circle is 0.7854 x D x D (that is pi / 4, and the diameter trap it guards against is covered in Circles in the Field: Area, Circumference and the Diameter Trap). Multiply by the height and you have cubic feet. A cubic foot holds 7.48 gallons, so 0.7854 x 7.48 = 5.875 gallons per foot of height per square foot of D squared. The 7.48 is owned by The Water Constants 7.48, 8.34 and 231.
Rectangular tank: gallons per inch of depth = L x W / 231, with the inside length L and width W in inches. Multiply by the water depth in inches for the total.
Where 231 comes from: it is the number of cubic inches in a US gallon, by definition. A slab of the tank 1 in deep holds L x W x 1 cubic inches, and dividing by 231 turns that into gallons. Working per inch of depth is deliberate, because it is the number you want when the question is "how far will the level drop if I pull 50 gallons" or "how many inches is one bag of salt dissolved into."
TANK VOLUME WORKSHEET (fill one per tank)
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VERTICAL ROUND RECTANGULAR
D inside ____ ft L inside ____ in
H water ____ ft W inside ____ in
D x D ____ sq ft L x W ____ sq in
x 5.875 / 231 = ____ gal/in
x H = ____ gal x depth ____ in
= ____ gal
CHECK: box it, then x 0.785 for round
(D x D x H x 7.48 x 0.785)
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Brim or working level? ____ Wall t: ____ in
The steps
1. Decide what level you are figuring to. Brim capacity, the overflow, the normal operating level and the current level are four different volumes in the same tank. A brine tank, a sump or a mix tank almost never runs full. Write on the sheet which one you want before you measure, because "the tank holds 264 gallons" and "there are 264 gallons in it" are not the same statement, and a dose figured on the first when the tank sits at two thirds is half again too strong (1 / 0.667 = 1.5).
2. Measure the inside, or measure the outside and subtract two walls. The formulas want the water's dimensions. On a thin steel or fiberglass tank the outside is close enough. On a molded plastic tank the wall can be thick enough to matter, and the error is squared on a round tank: a 36 in outside diameter with a 1/2 in wall is 35 in inside, and (35 / 36) x (35 / 36) = 0.945, so the outside figure overstates the volume by about 5.8 percent (1 / 0.945 = 1.058). Measure wall thickness at a fitting opening, or read it off the tank's data plate.
3. Measure height from the inside bottom to the level you chose. Tanks with a raised or domed bottom, a cone bottom or a sump section are not cylinders all the way down. Measure the straight-wall section and treat the odd bottom as its own piece, or take the maker's figure for it. If you cannot see the bottom, a stick or tape lowered to touch it, marked at the rim, and compared to the outside height tells you how much of the outside height is actually bottom structure.
Hazard at this step: lowering a tape into a tank is fine; putting your head or body into the opening is not. A closed tank can hold an atmosphere that cannot support life (a septic, sewage, fuel or treated-water tank especially). Read from outside the opening, and treat any entry as a permit-required confined space under 29 CFR 1910.146, which is not this procedure. On a chemical tank, read the product's safety data sheet for its inhalation and splash hazards before the lid comes off, and wear the eye and hand protection it names.
4. Convert to one unit before multiplying. The round formula wants feet; the rectangular one wants inches. Mixed inputs are the commonest slip, and it does not produce a small error: one dimension in inches and one in feet throws the answer off by a factor of 12. Write the conversion as its own line. See related: Units Carried Through Every Line of a Calculation.
5. Run the formula, then run the box check. The check costs ten seconds. Imagine the tank in the smallest box it fits in, figure that box in gallons, and take 0.785 of it for a round tank. If the box check and the formula disagree by more than rounding, one of them has a dimension in the wrong unit or a radius where a diameter belongs.
6. Round only at the end, and round a dose toward caution. Carry the unrounded volume to the dose arithmetic. Then round in the direction the job needs: for a chemical that is dangerous in excess, figure from the lower volume and add more after a test, never the reverse. See related: Write the Calculation Down: The Field Math Habit.
Worked example 1: a vertical round tank, 3 ft by 5 ft
A vertical tank measures 3 ft inside diameter and the water stands 5 ft deep.
Line Item Value Unit
1 D inside 3.00 ft
2 D x D 9.00 sq ft
3 x 5.875 52.875 gal per ft of H
4 H water 5.00 ft
5 Volume = L3 x L4 264.4 gal
6 Box: 3 x 3 x 5 45.0 cu ft
7 L6 x 7.48 336.6 gal
8 L7 x 0.785 264.2 gal
The tank holds about 264 gallons at that level. The box check lands at 264.2 against 264.4, and the 0.2 gallon gap is only the rounding in 0.785 (pi / 4 is 0.7854). If line 5 had come out near 1,060, someone used the diameter as a radius in a pi x r x r formula, which multiplies by four; off by a factor of 12 or 144, someone mixed inches with feet.
Level instead of brim. If the same tank is 6 ft tall and the dose is for the 5 ft operating level, the brim capacity of 5.875 x 9 x 6 = 317.3 gallons is 20 percent more than the water actually in it (317.3 / 264.4 = 1.20). That 20 percent goes straight into the dose.
Worked example 2: a rectangular tank, by the inch
A rectangular tank measures 60 in by 36 in inside, with the water 30 in deep.
Line Item Value Unit
1 L x W = 60 x 36 2,160 sq in
2 L1 / 231 9.351 gal per in
3 Depth 30 in
4 Volume = L2 x L3 280.5 gal
5 Check in feet:
5 x 3 x 2.5 37.5 cu ft
6 L5 x 7.48 280.5 gal
280.5 gallons, and the feet route agrees exactly. The per-inch figure on line 2 is the one to write on the tank with a marker: 9.35 gallons per inch. Pull 47 gallons out and the level falls about 5 in (47 / 9.351 = 5.03), which lets the next tech read the drawdown with a tape instead of a calculator.
The horizontal round tank: why a straight line lies
A round tank lying on its side has the same full volume formula (5.875 x D x D x the length). What changes is the partial fill. The tank is narrow at the bottom, widest at the middle and narrow again at the top, so each inch of depth near the bottom holds far less than an inch at the centerline. Reading the stick as a straight fraction of the diameter overstates the contents low in the tank and understates them high.
How far off, for a plain cylinder with flat ends (fraction of the full volume at a fraction of the diameter):
DEPTH AS FRACTION VOLUME AS FRACTION
OF DIAMETER OF FULL (cylinder)
--------------- ------------------
0.10 0.052
0.25 0.196
0.50 0.500
0.75 0.804
0.90 0.948
Take a horizontal tank 4 ft in diameter and 8 ft long. Full, it holds 5.875 x 16 x 8 = 752 gallons. The stick reads 12 in, a quarter of the 48 in diameter. A straight line says 752 x 0.25 = 188 gallons. The geometry says 752 x 0.196 = 147 gallons, so the straight line claims 41 gallons that are not there (188 minus 147), which is 28 percent too many (188 / 147 = 1.28). On a fuel tank that is the difference between a delivery scheduled in time and a customer who runs out.
That table is still not the answer for a real tank. It is exact only for a perfect cylinder with flat ends. Real horizontal tanks have dished or rounded heads that hold extra volume, many residential fuel tanks are oval or flat-sided rather than round, and some sit tilted for drainage so the stick reading depends on which end you measure. Every one of those breaks the table. The maker publishes a gauging chart (inches on the stick to gallons) for that exact tank model, often on the tank label or in the installation literature, and that chart is what you read. Use the table above only to recognize that a straight-line reading is wrong, and by roughly how much.
Why the middle is the one safe point. At exactly half the diameter the cylinder is half full whatever its shape, because the top half mirrors the bottom. That is a useful self-check on a maker's chart: its half-diameter row should read half the full capacity.
What changes the answer
- The tank is not upright or not round. A slanted vertical tank, an oval tank or a tank with a cone bottom needs its own geometry or the maker's figure. Do not stretch the round formula over a shape it was not derived for.
- Something inside takes up room. A heating coil, a media bed, a float assembly or a pump displaces water. For a dose on a treatment tank, the maker's stated water volume beats your geometry, because it subtracts what you cannot see.
- The liquid is not water. The gallons are still gallons, but the 8.34 pounds per gallon that turns volume into weight does not hold for brine, fuel or glycol mix. See related: How to Tally the Load of a Filled Tub or Tank on a Floor.
How to verify you got this right
- The box check agrees with the formula to within the rounding of 0.785.
- Every dimension on the sheet says inside or outside, and every line carries a unit.
- The level is named. The sheet says brim, operating or current, and the dose used the one the tank is actually at.
- A horizontal tank's partial reading came from the maker's chart for that model, and the chart's half-diameter row reads half of full.
- One observed drawdown matches the per-inch figure. After a known transfer, the level change in inches times the gallons per inch should come close to what you moved. If it does not, a dimension is wrong.
References
- See related: The Water Constants 7.48, 8.34 and 231
- See related: Circles in the Field: Area, Circumference and the Diameter Trap
- See related: Units Carried Through Every Line of a Calculation
- 29 CFR 1910.146, Permit-Required Confined Spaces (OSHA)
- Tank manufacturer gauging charts and data plates