Sphere Tank Volume Calculator: How to Find Capacity

This calculator works out the volume of a spherical tank (a sphere-shaped vessel) in liters, gallons, and cubic meters. You enter the tank’s inside diameter and, if you want a partial fill, the liquid depth; it returns the total capacity and the current liquid volume. The result is based on the standard geometric formula for a sphere and a spherical cap, so it is accurate for any sphere where you can measure the internal diameter.

Formula

The total volume of a sphere is given by the classic formula:

V_total = (4/3) × π × r³

where r is the radius (half the diameter) and π ≈ 3.14159. If you measure the diameter D, the formula becomes V = (π × D³) / 6. The result is in cubic units of whatever you used for the diameter (e.g., cubic meters, cubic feet).

For a partially filled sphere (liquid depth h from the bottom), the liquid forms a spherical cap. Its volume is:

V_cap = (π × h² × (3r − h)) / 3

where h is the liquid depth and r is the sphere radius, both in the same units. This formula works for any depth from 0 up to the full diameter (2r). When h = 2r, the cap volume equals the full sphere volume.

Worked Example

Using the calculator defaults: a spherical tank with diameter 2 m, filled to a depth of 1 m.

1. Radius: r = d/2 = 2/2 = 1 m
2. Total volume: V = (4/3) × π × (1)³ = (4/3) × 3.14159 × 1 = 4.18879 m³
3. Convert to liters: 4.18879 m³ × 1000 = 4188.79 L → 4188.8 L (as shown)
4. Liquid volume: V_cap = (π × 1² × (3×1 − 1)) / 3 = (π × 1 × 2) / 3 = 2.0944 m³ = 2094.4 L
5. Fill percentage: (2094.4 / 4188.8) × 100 = 50.0%

So a 2 m diameter sphere filled halfway (1 m depth) holds exactly half its capacity: about 2094 liters. The calculator shows 4188.8 L total and 2094.4 L liquid, which matches.

When to Use This Calculator

Spherical tanks are common in industrial storage (propane, chemicals, food products), water treatment, and even home rainwater systems. You might need the volume for:

• Estimating how much liquid a tank can hold before you buy or install it.
• Determining the current contents when you only know the liquid depth (e.g., from a sight glass or dipstick).
• Calculating the volume of a spherical pressure vessel or a decorative sphere (like a garden water feature).
• Converting between metric and imperial units for shipping, chemical dosing, or regulatory reporting.

The calculator is especially useful when you have a partially filled tank: the spherical cap formula is not as widely known as the full sphere formula, and a quick calculation avoids guesswork.

How to Measure a Spherical Tank

Accuracy starts with the measurement. For a real tank, you need the internal diameter – not the outside diameter, because the wall thickness reduces the volume. If you only have the outside diameter, subtract twice the wall thickness. For example, a tank with an outside diameter of 1.5 m and 5 mm thick walls has an internal diameter of 1.5 − 0.010 = 1.49 m.

For the liquid depth, measure from the lowest point inside the tank (the bottom) to the liquid surface. If the tank is horizontal and has a dipstick, the depth is the vertical distance from the bottom. If you are using a level sensor, make sure it is calibrated to the bottom of the tank, not the center.

When the tank is not a perfect sphere (e.g., it has flat ends or a standpipe), the formula will slightly over- or under-estimate. For most industrial spheres, the deviation is small, but for exact inventory you may need a calibration table from the manufacturer.

Common Mistakes and How to Avoid Them

• Unit mix-ups: If you measure the diameter in feet, the volume is in cubic feet. To get gallons, multiply by 7.48052 (US gallons per cubic foot). To get liters, multiply by 28.3168. The calculator above uses meters and liters; if you have feet, convert first (1 ft = 0.3048 m).
• Using the diameter instead of the radius: The sphere formula uses radius cubed. If you accidentally use the diameter as the radius, the volume becomes 8 times too large. Always halve the diameter first.
• Forgetting to account for wall thickness: The volume is based on the inside dimensions. A 2 m outside diameter tank with 10 mm walls holds about 1.5% less than a 2 m inside diameter tank.
• Rounding too early: If you round π to 3.14 and the radius to one decimal, you can get a 2% error. Keep extra digits until the final answer.
• Misinterpreting the liquid depth: The spherical cap formula assumes the depth is measured vertically from the bottom. If the tank is tilted, the formula is not accurate.

Limitations of the Formula

The formulas above are exact for a perfect mathematical sphere. Real tanks may have:

• Manhole openings, nozzles, or internal structures that displace liquid.
• Elliptical or torispherical heads (common in pressure vessels) – these are not spheres and require different formulas.
• Thermal expansion: liquids expand with temperature, so the volume at 20°C differs from that at 80°C. For water, the change is about 2% over a 60°C range.
• Vapor space: if the tank is not completely filled, the liquid volume is what you calculate, but the total capacity includes the vapor space above.

For highly accurate inventory, you would need a strapping table (a calibration chart) that accounts for the tank’s actual geometry. But for most purposes, the sphere formula is within a few percent.

Practical Examples

Let’s look at two common scenarios:

Example 1: Full tank – A spherical tank has an internal diameter of 3 m. What is its capacity in gallons?
Radius = 1.5 m. Volume = (4/3) × π × (1.5)³ = 14.137 m³. Convert to liters: 14,137 L. Convert to US gallons: 14,137 / 3.78541 = 3735 gallons. (The calculator would show 14,137.0 L and 3735 gal if you added a gallon output.)

Example 2: Partial fill – The same 3 m tank has a liquid depth of 1.2 m. What is the liquid volume?
h = 1.2 m, r = 1.5 m. V = (π × 1.2² × (3×1.5 − 1.2)) / 3 = (3.14159 × 1.44 × 3.3) / 3 = 4.974 m³ = 4974 L. Fill percentage = 4974 / 14137 = 35.2%.

These examples show how the same tank can be used for different volumes depending on the liquid level.

Frequently Asked Questions

How do I find the volume of a spherical tank in gallons?
Measure the internal diameter in feet, divide by 2 to get the radius, then use V = (4/3)πr³ to get cubic feet. Multiply by 7.48052 to get US gallons. For example, a 4 ft diameter sphere (r=2 ft) has V = 33.51 ft³, which is 250.7 gallons.

What is the formula for a half-full sphere?
When the liquid depth is exactly half the diameter (h = r), the liquid volume is half the total volume. You can use the cap formula with h = r: V = (π × r² × (3r − r)) / 3 = (2πr³)/3, which is indeed half of (4πr³)/3.

Can I use this for a propane tank?
Yes, if the tank is a true sphere. Many propane tanks are spherical (often called “bullets” when horizontal). However, you must use the internal diameter and account for the fact that propane is stored as a liquid under pressure. The volume you calculate is the total liquid capacity; safety regulations typically limit filling to 80-85% of that volume to allow for expansion.

Why does the calculator give liters when I measured in meters?
Because 1 cubic meter equals 1000 liters. The calculator converts the cubic meter result to liters by multiplying by 1000. If you measure in centimeters, you would get cubic centimeters, which are the same as milliliters (1 mL = 1 cm³).

How accurate is the spherical cap formula for a partially filled tank?
It is mathematically exact for a perfect sphere. The only inaccuracies come from measurement errors (diameter, depth, wall thickness) and deviations from a perfect sphere. For a well-maintained tank, the formula is accurate to within a few percent.

What if my tank is not exactly spherical?
If the tank has elliptical ends or a cylindrical section, use the appropriate formulas for those shapes. For a horizontal cylinder with spherical ends, you would need to combine the cylinder volume and the two cap volumes. This calculator is only for a pure sphere.

Related

• Cubic Meter to Square Feet Calculator
• Cone Bottom Tank Volume Calculator
• Conical Bottom Tank Volume Calculator & Formula

in d | Tank diameter | m | 2 | 0
in h | Liquid depth | m | 1 | 0
out total | Total capacity | L | (4/3) * pi * (d/2)^3 * 1000 | 1
out liquid | Liquid volume | L | (pi * h^2 * ((3 * d / 2) - h)) * 1000 | 1
out pct | Fill percentage | % | (liquid / total) * 100 | 1