Capsule Calculator
Volume and surface area of a capsule (pill) shape.
Volume
395.8407
cylinder + 2 hemispheres
Surface area
301.5929
Total length
16
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Capsule Calculator works
The Capsule Calculator instantly computes the volume and total surface area of a capsule shape — the classic pill or stadium-of-revolution geometry formed by a cylinder capped with two hemispheres — making it essential for engineers, pharmacists, designers, and students working with 3D geometry.
A capsule is a specific 3D solid defined by exactly two measurements: the radius of the circular cross-section and the height of the cylindrical body (the straight middle section, not the total end-to-end length). The two hemispherical end-caps together form one complete sphere of the same radius. This means the capsule is essentially a cylinder with a sphere seamlessly merged onto both ends, creating the smooth, rounded shape familiar in pharmaceutical pills, food containers, and aerospace fuel tanks.
The volume calculation breaks the shape into two components: the central cylinder and the combined sphere formed by the two end-caps. The cylindrical portion has volume π·r²·h, where r is the radius and h is the length of the straight cylindrical section. The two hemispheres together contribute (4/3)·π·r³, the standard sphere volume formula. Adding these gives the total capsule volume. Because the hemispherical caps are already accounted for within the sphere term, there is no double-counting at the joints — the geometry fits together perfectly.
Surface area is calculated similarly by summing the lateral surface of the cylinder (2·π·r·h) and the full surface area of the combined sphere (4·π·r²). Notice that the flat circular ends of the cylinder are not included in the surface area — they are replaced by the hemispherical caps, so the formula naturally excludes any flat disc areas. A common mistake is entering the total end-to-end length of the capsule as 'h' rather than just the cylindrical body height; doing so significantly overestimates both volume and surface area. Always measure h as the distance between the two flat points where the hemisphere meets the cylinder.
Understanding how each dimension affects the result is critical for practical applications. Volume scales with the square of the radius (r²) in the cylinder term and the cube of the radius (r³) in the sphere term, meaning even a small increase in radius dramatically increases volume — a 10% increase in radius can raise volume by over 20%. Height, by contrast, has a strictly linear effect on the cylindrical volume component only. For pill design or container optimization, this means adjusting radius is a far more powerful lever for changing capacity than adjusting the cylindrical length.
Formula
V = πr²L + 4/3·πr³
Pro tips
- Always input the cylindrical body height (the straight middle section), not the overall pill length. If you only know the total length, subtract 2·r (one full diameter) to get the correct h value.
- If you need to scale a capsule design to double its volume, you cannot simply double just one dimension — because radius has a cubic relationship with the sphere component, increasing r by about 26% while keeping h constant will roughly double the spherical volume contribution. Use the calculator iteratively to dial in your target.
- For pharmaceutical applications, volume in milliliters (mL) equals volume in cubic centimeters (cm³), so inputting radius and height in centimeters gives you fill volume directly in mL — a useful shortcut for capsule sizing.
- Surface area is proportional to coating or material cost in manufacturing. If minimizing surface area for a fixed volume is your goal, a capsule with h = 0 (a pure sphere) is the most efficient shape; any nonzero cylindrical height increases surface-area-to-volume ratio.
- When measuring a physical capsule, use digital calipers for accurate radius and length readings, and remember that the capsule shell itself has wall thickness — if you need the interior (fill) volume, subtract the shell thickness from r before calculating.
Key terms
- Radius (r)
- — The radius of the circular cross-section of the capsule, which is uniform across both the cylindrical body and the hemispherical end-caps.
- Cylindrical Height (h)
- — The length of the straight, cylindrical middle section of the capsule, measured between the two points where the hemisphere caps begin — not the total end-to-end length of the shape.
- Hemisphere
- — Exactly half of a sphere; a capsule has two hemispheres — one at each end — which together form one complete sphere of the same radius as the cylinder.
- Lateral Surface Area
- — The curved outer surface of the cylindrical portion of the capsule, equal to 2·π·r·h, excluding the hemispherical caps.
- Stadium of Revolution
- — The 2D 'stadium' shape (a rectangle with semicircular ends) rotated around its central axis to produce the 3D capsule solid.
- Total Surface Area
- — The complete outer surface of the capsule, combining the lateral cylinder surface and the full surface area of the sphere formed by both hemispherical caps.
