Capsule Surface Area Calculator (Pill Shape)

A capsule (spherocylinder) is a cylinder with hemispherical caps on each end. Surface area combines the cylinder side with one full sphere (the two hemispheres make one sphere).

SA = 2 × π × r × h + 4 × π × r² = 2 × π × r × (h + 2r)

Where r is the radius (also the cap hemisphere radius) and h is the cylinder length (NOT counting the rounded ends).

Worked example: painting a 100 lb propane tank A typical 100 lb propane tank: r = 6 in, total length 48 in (so cylinder length h = 48 − 2 × 6 = 36 in). SA = 2π × 6 × (36 + 12) = 12π × 48 = 576π ≈ 1,810 sq in = 12.6 sq ft.

Industrial-grade tank paint covers about 50-100 sq ft per gallon depending on viscosity. A gallon of marine-grade epoxy paint covers ~70 sq ft per coat. Two coats on this tank uses about 1/3 gallon, plus prep work and mineral spirits for cleanup.

Worked example: pharmaceutical pill coating A size 0 vitamin capsule: r = 3.4 mm, total length 21.7 mm (so cylinder length h = 21.7 − 6.8 = 14.9 mm). SA = 2π × 3.4 × (14.9 + 6.8) = 6.8π × 21.7 ≈ 463.4 mm² per capsule.

For a batch of 10,000 capsules: 4.63 m² of total surface, which is 46,340 cm².

A pan-coating tumbler lays down roughly 5-10 mg of dried coating per cm², depending on whether it is a cosmetic film or a functional enteric layer. That is about 23-46 mg on each capsule, and 230 to 460 grams for the whole batch. Worth doing the multiplication rather than eyeballing it: coating material is priced by the kilo, and being out by a factor of ten on a production run is an expensive way to learn arithmetic.

Where capsule surface area matters in practice:

• Pill coating chemistry. Time-release coatings, flavor masking, enteric coatings are all priced per surface area.
• Propane and LPG tank painting. White or silver “all-weather” tank paint.
• Pressure vessel external coating. Industrial chemical storage, fuel tanks, air receivers.
• Submarine and torpedo hulls. Anti-fouling and anti-corrosion paint.
• Vitamin and supplement capsule shells. Gelatin or vegetable-based shell material.
• Buoys and floats. Marine navigation buoys with pill-shape designs.

Why pressure vessels are shaped this way

For a thin wall under internal pressure, the hoop stress in a cylinder is pr/t, while the stress in a spherical shell of the same radius and thickness is pr/2t. A hemisphere carries exactly half the stress of the cylinder it caps. Flat ends are worse still, because a flat plate resists pressure in bending rather than in tension, and bending stress climbs with the square of the diameter.

That is why almost every pressure vessel you will ever see, propane, oxygen, nitrogen, compressed air, has domed ends. The head is never the part that fails.

Comparing capsule SA to a cylinder of the same total length:

A cylinder of length (h + 2r) and radius r: SA_cyl = 2πr × (h + 2r) + 2πr² (lateral + 2 flat ends) SA_capsule = 2πr × h + 4πr² (lateral + sphere)

Subtract: SA_cyl − SA_capsule = 2πrh + 4πr² + 2πr² − 2πrh − 4πr² = 2πr²

The difference comes out positive, so the flat-ended cylinder has 2πr² more surface, not less. Rounding the ends does not cost material, it saves it.

That surprises people, and it should not. A sphere encloses the most volume for the least surface of any shape there is, so bending a flat end into a hemisphere always trades surface away. Check it on the propane tank above: the capsule works out at 1,810 sq in, while a flat-ended cylinder 48 in long of the same radius needs 2,036 sq in. That is 226 sq in more paint, or 2πr² = 2π × 36, exactly as the algebra says.

So the hemispherical end caps win twice over: less wall stress and less material. What you give up is interior volume, because the same 48 inches of length holds (2/3)πr³ less once you round the ends off. Strength and surface for capacity, and for a pressure vessel that is an easy trade.

Sanity check:

• h = 0: capsule = sphere. SA = 4πr². ✓
• r → 0: capsule → line. SA → 0. ✓
• h = r case: SA = 2πr × 3r = 6πr².