Quick answer: a capsule holds V = πr²a + (4/3)πr³ — the cylinder plus both hemispherical caps, which together equal one full sphere. Enter the cylinder section only; if you measured tip to tip, subtract both caps first: a = L − 2r.
The capsule formula, step by step
where r = radius · a = length of the straight cylindrical section · overall length L = a + 2r
The elegance here is that two hemispheres make one sphere, so the shape decomposes into exactly two familiar pieces. Worked example — a capsule with a 2-unit radius and a 5-unit cylinder section:
- Cylinder: π × 2² × 5 = 62.83
- Both caps (one sphere): (4/3) × π × 2³ = 33.51
- Total: 62.83 + 33.51 = 96.34 cubic units
If you only know the overall tip-to-tip length, convert first. With L = 9 and r = 2, the cylinder section is 9 − 4 = 5 — exactly the case above. Forgetting this subtraction is the single most common error on this shape, and it always overestimates.
Horizontal propane tanks — nominal sizes
Domestic propane tanks are capsules, and their listed capacity is the tank's full water volume; regulations limit the actual propane fill to about 80% to leave room for thermal expansion:
| Nominal size | Diameter | Overall length | Water volume | 80% fill |
|---|---|---|---|---|
| 120 gal (vertical) | 24 in | 4 ft 6 in | ≈ 120 gal | 96 gal |
| 250 gal | 30 in | 7 ft 10 in | ≈ 250 gal | 200 gal |
| 500 gal | 37 in | 9 ft 11 in | ≈ 500 gal | 400 gal |
| 1,000 gal | 41 in | 16 ft 1 in | ≈ 1,000 gal | 800 gal |
Try the 500-gallon row in the calculator: r = 18.5 in, cylinder = 119 − 37 = 82 in. The result lands within a couple of percent of the nameplate — the small gap comes from real tanks using shallower dished heads rather than true hemispheres.
Where capsule volume shows up in real life
Pressure vessels and gas storage
Rounded ends are an engineering necessity, not a style choice. A flat plate under internal pressure bends; a hemisphere carries the load in pure tension and can be roughly half the thickness for the same rating. That is why every propane tank, air receiver and scuba cylinder is a capsule rather than a drum. Working out how much is left in one is a different problem from total capacity — a partly filled horizontal vessel needs circular-segment maths, which the fuel tank volume calculator applies to a dipstick reading.
Pharmaceutical capsules
Two-piece gelatin capsules are sized 000 down to 5. A size 0 — the most common for supplements — is about 7.34 mm across with a 21.7 mm closed length, giving roughly 0.68 mL of fill space. Formulators work backwards from that volume and the powder's bulk density to decide what dose actually fits.
Tanker trailers and rail cars
Road tankers are capsules lying on their side. A trailer with a 6-ft diameter and 40-ft overall length holds π × 9 × 34 + (4/3)π × 27 ≈ 1,074 ft³ ≈ 8,030 gallons. For partially filled horizontal tanks, the fill-level maths is genuinely awkward — the tank volume calculator handles that case properly.
Common mistakes & pro tips
- Overall length used as the cylinder length. Always subtract both caps: a = L − 2r. On a short, fat capsule this error can inflate the result by 30% or more.
- Diameter entered as radius. Tanks and pills are both specified by diameter. Use Diameter mode rather than halving in your head.
- Assuming hemispherical heads. Many real tanks use elliptical or torispherical (dished) heads that are shallower than a true hemisphere, so the true volume runs slightly under this calculator's figure. For nameplate-critical work, use the manufacturer's chart.
- Pro tip — usable vs. total. Gas tanks are filled to about 80% and liquid tanks are rarely brim-full either. Compute the geometric volume here, then apply the fill fraction your application requires.
Frequently asked questions
What is the formula for the volume of a capsule?
V = πr²a + (4/3)πr³ — the straight cylinder section plus the two hemispherical caps, which together make one complete sphere. A capsule with a 2-unit radius and 5-unit cylinder length holds π × 4 × 5 + (4/3)π × 8 ≈ 96.3 cubic units.
Do I measure the total length or just the cylinder?
This calculator wants the straight cylindrical section only. If you measured the overall length L from tip to tip, subtract both caps: a = L − 2r. A propane tank 10 ft overall with a 2-ft radius has a cylinder section of 10 − 4 = 6 ft.
How do I calculate the capacity of a propane tank?
Most horizontal propane tanks are capsules. A 500-gallon tank is typically about 37 in in diameter and 10 ft overall: r = 1.54 ft, cylinder = 10 − 3.08 = 6.92 ft, giving π × 2.37 × 6.92 + (4/3)π × 3.66 ≈ 66.9 ft³ ≈ 500 gallons. Note that propane tanks are only filled to about 80% to allow for thermal expansion.
Are pill capsules really this shape?
Yes — a two-piece gelatin capsule is a cylinder with hemispherical ends, which is where the shape gets its name. A size 0 capsule is about 7.34 mm in diameter with a 21.7 mm overall length, giving roughly 0.68 mL of fill volume.
What is the difference between a capsule and a cylinder?
A cylinder has flat ends; a capsule has hemispherical domes on both. Those two domes add exactly one sphere's worth of volume, (4/3)πr³. Pressure vessels use rounded ends because curved surfaces handle internal pressure far better than flat plates, which would need to be much thicker.