Horizontal tank volume calculator
This horizontal tank volume calculator gives the liquid in a cylinder lying on its side at any depth. The liquid's cross-section is a circular segment of area A = r² acos((r − f) ÷ r) − (r − f) √(2rf − f²), where r is the radius and f the liquid depth. Multiply A by the tank length to get the volume.
Contents
Horizontal tank 48 in across and 96 in long, 12 in of liquid: A = 24² acos(12 ÷ 24) − 12 √(2 × 24 × 12 − 12²) = 353.77 in². V = 353.77 × 96 = 33,962 in³ = 147.0 US gal, out of 752.0 gal when full.
The calculator above takes the inside diameter, the length and the liquid depth. It returns the volume of liquid, the share of the tank that is filled and how much room is left, and its Chart tab lists the volume at every depth. Set it to a vertical tank to compare.
Why a horizontal tank needs a different formula
When a cylinder is full, its orientation does not matter: the volume is π × r² × L either way, as on the cylinder tank volume page. The difference appears when the tank is partly full.
In a vertical tank, each inch of depth adds a full circle of liquid, so the volume rises in a straight line with the depth. In a horizontal tank, each inch of depth adds a thin strip whose width is the width of the liquid surface. That width is 2√(2rf − f²). For a tank 48 in across, it is only 13.7 in at 1 in of depth, 41.6 in at 12 in, and 48 in at the middle. So the bottom inch of a 48 in × 96 in tank holds about 3.8 gallons, while the inch around the middle holds about 19.9 gallons. The gauge is not linear.
The partially filled horizontal cylinder formula
The liquid fills a circular segment: the part of the circle below a horizontal chord. Its area is
A = r² × acos((r − f) ÷ r) − (r − f) × √(2rf − f²)
and the volume of liquid is V = A × L.
- r is the inside radius, half the inside diameter.
- f is the liquid depth, from the bottom of the tank to the surface, between 0 and 2r.
- L is the length of the cylindrical shell.
- acos must return an angle in radians. A calculator in degree mode gives a wrong answer.
The first term, r² × acos((r − f) ÷ r), is the area of the circular sector that holds the liquid. The second term removes the triangle between the center and the surface chord. When the tank is more than half full, r − f is negative and the second term becomes an addition; the same formula works from empty to full without a separate case.
An equivalent form uses the central angle θ = 2 acos((r − f) ÷ r): A = r² (θ − sin θ) ÷ 2. Both give the same area. For the half cylinder, at f = r, either one gives πr² ÷ 2, as on the page about half cylinder volume.
Worked example: a 48 in × 96 in tank
A horizontal tank has an inside diameter of 48 in and a length of 96 in. The dipstick shows 12 in of liquid.
- Radiusr = 48 ÷ 2 = 24 in. Then r − f = 24 − 12 = 12 in.
- Angleacos(12 ÷ 24) = acos(0.5) = 1.0472 rad (60°).
- Sectorr² × 1.0472 = 576 × 1.0472 = 603.19 in².
- Triangle√(2 × 24 × 12 − 12²) = √432 = 20.785; 12 × 20.785 = 249.42 in².
- SegmentA = 603.19 − 249.42 = 353.77 in².
- Volume353.77 × 96 = 33,962 in³ ÷ 231 = 147.0 US gallons (556.5 L).
The full tank holds π × 24² × 96 = 173,717.5 in³ = 752.0 gallons. So 12 in of depth, a quarter of the diameter, is 19.6 % of the capacity. Read as if the tank were linear, a quarter of 752 is 188.0 gallons, which overstates the contents by 41 gallons.
At 36 in of depth, r − f = −12. acos(−0.5) = 2.0944 rad, the sector is 1,206.37 in², and the triangle term becomes + 249.42 in², for A = 1,455.79 in². The volume is 1,455.79 × 96 ÷ 231 = 605.0 gallons, or 80.4 % of the tank.
Metric example
A tank 1.5 m across and 3 m long holds 0.5 m of liquid. r = 0.75 m and r − f = 0.25 m. acos(0.25 ÷ 0.75) = 1.23096 rad, so the sector is 0.5625 × 1.23096 = 0.69241 m². √(2 × 0.75 × 0.5 − 0.5²) = √0.5 = 0.70711, and 0.25 × 0.70711 = 0.17678 m². A = 0.51564 m², and V = 0.51564 × 3 = 1.5469 m³, or 1,546.9 liters. Full, the tank holds 5,301.4 L, so it is 29.2 % full at a third of its depth.
Depth percentage to volume percentage
Because the shape of the segment only depends on the ratio f ÷ d, the table below works for every horizontal cylinder with flat ends. Find the depth as a share of the inside diameter, then multiply the full capacity by the share of volume.
| Depth (% of diameter) | Volume (% of full) | Depth (% of diameter) | Volume (% of full) |
|---|---|---|---|
| 5 % | 1.87 % | 55 % | 56.36 % |
| 10 % | 5.20 % | 60 % | 62.65 % |
| 15 % | 9.41 % | 65 % | 68.81 % |
| 20 % | 14.24 % | 70 % | 74.77 % |
| 25 % | 19.55 % | 75 % | 80.45 % |
| 30 % | 25.23 % | 80 % | 85.76 % |
| 35 % | 31.19 % | 85 % | 90.59 % |
| 40 % | 37.35 % | 90 % | 94.80 % |
| 45 % | 43.64 % | 95 % | 98.13 % |
| 50 % | 50.00 % | 100 % | 100 % |
The table is symmetric: the volume at 70 % of the depth (74.77 %) plus the volume at 30 % (25.23 %) makes 100 %. Example: a 1,000-gallon horizontal tank with a 64 in diameter, filled to 16 in (25 %), holds about 1,000 × 0.1955 = 195.5 gallons. For a full depth-by-depth table of one tank, made for a dipstick, use the tank dip chart generator.
Vertical tank volume calculator: the linear case
Switch the calculator to vertical and the liquid volume is simply π × r² × f. A standing tank 48 in across and 96 in tall, filled to 24 in, holds a quarter of its 752.0 gallons: 188.0 gallons. Each inch of depth is worth 7.834 gallons from the bottom to the top. A horizontal tank of the same size, filled to the same 24 in, holds 376.0 gallons, because 24 in is half its diameter.
Tank ends and other limits
The formula assumes flat ends and a level tank. Three situations change the result:
- Rounded ends. A capsule-shaped tank with hemispherical ends holds extra liquid in the heads. Compute the shell with this formula and the heads separately: the page on the cylinder with hemisphere volume covers them. Dished heads that are shallower than a hemisphere need the manufacturer's figure or a calibration by filling.
- A tilted tank. If the tank is not level, the depth at the dip point is not the depth everywhere. Measure at the center of the length, and level the tank if you need an accurate gauge.
- Outside dimensions. Use the inside diameter. On a steel tank, subtract twice the wall thickness from the outside measurement; for a pipe, the pipe volume calculator works from the inside diameter in the same way.
Common mistakes
- acos in degrees. acos(0.5) is 60 in degrees but 1.0472 in radians. The formula needs radians.
- Depth measured from the top. If you measured the empty space, the depth is the diameter minus that gap.
- Diameter for radius. r is half the diameter, and r² appears in the formula.
- Mixed units. Put r, f and L in the same unit, then convert the result: 231 in³ per US gallon, 1,000 cm³ per liter. The basic cylinder volume calculator lists the factors.
Quick check: whatever the tank, half the depth is half the volume. If your calculation gives anything else at f = r, the formula was entered wrongly.
FAQ
What is the formula for the volume of a partially filled horizontal cylinder?
V = L × [r² acos((r − f) ÷ r) − (r − f) √(2rf − f²)], where r is the inside radius, f the liquid depth and L the length. The bracket is the area of the circular segment covered by the liquid. The angle from acos must be in radians.
Why is half the depth not half the volume in every case?
At exactly half the depth it is: the liquid fills half the circle. At any other level it is not, because the liquid surface is narrow near the bottom and top and widest at the middle. A quarter of the depth holds only 19.55 % of the volume.
How do I measure the liquid in a horizontal cylinder tank?
Put a straight stick down through a hole near the middle of the top until it touches the bottom, then read the wet mark: that is the depth f. Enter it with the inside diameter and the length in the calculator, or look it up in a dip chart for your tank.
How do I calculate this in Excel?
With the radius in A2, the depth in B2 and the length in C2: =C2*(A2^2*ACOS((A2-B2)/A2)-(A2-B2)*SQRT(2*A2*B2-B2^2)). Excel's ACOS returns radians, which is what the formula needs. Divide by 231 for US gallons if the lengths are in inches.
Is the volume of a full horizontal tank different from a vertical one?
No. A full cylinder holds π × r² × L in any position. Only the partly filled volume depends on whether the tank stands up or lies down.
Does the formula work for tanks with rounded ends?
It covers the cylindrical shell only, with flat ends. Hemispherical or dished heads hold extra liquid that must be added separately; the shell result alone underestimates such a tank.
Related
- Cylinder tank volumeTank volume calculator for vertical and horizontal cylinder tanks, full or partly filled. Get the capacity in gallons and liters, with worked examples.
- Tank dip chartMake a tank dip chart for any cylindrical tank: depth to gallons and liters, a full worked chart for a 48 in × 72 in tank, and how to read a dipstick.
- Half cylinderA half cylinder holds V = πr²h ÷ 2. Get its volume, surface area and net, plus quarter cylinders and any sector, with worked examples in inches and cm.
Updated October 12, 2026