1,308.9969
units³
628.3185
units²
20
units
785.3982
units³
523.5988
units³
314.1593
units²
314.1593
units²
2
1,308.9969
units³
628.3185
units²
20
units
785.3982
units³
523.5988
units³
314.1593
units²
314.1593
units²
2
A capsule (also known as a stadium solid or discorectangle solid) is a three-dimensional shape consisting of a cylinder with two hemispherical caps on each end. Together, the two hemispheres form a complete sphere. This smooth, rounded shape is one of the most common geometric forms in everyday life, found in pharmaceutical capsules, fuel tanks, pressure vessels, stadium roofs, and even the shape of certain bacteria (bacilli).
The Capsule Calculator computes the total volume, surface area, total length, and the volume breakdown between the cylindrical and spherical portions. You need only two measurements: the radius $$r$$ and the cylinder height $$h$$ (the length of the straight cylindrical section, not including the hemispherical caps).
The volume of a capsule combines the cylinder volume and the sphere volume (two hemispheres = one sphere):
$$V = \pi r^2 h + \frac{4}{3}\pi r^3$$
This can be factored as:
$$V = \pi r^2 \left(h + \frac{4r}{3}\right)$$
The surface area is elegantly simple. The curved surface of the cylinder is $$2\pi r h$$, and the two hemispherical caps together form a complete sphere with surface area $$4\pi r^2$$:
$$A = 2\pi r h + 4\pi r^2 = 2\pi r(h + 2r)$$
Notice that the surface area formula has no flat circular areas — the hemispheres seamlessly continue where the cylinder ends, creating a smooth, continuous surface with no edges or seams.
The total length (the overall end-to-end dimension) is simply:
$$L = h + 2r$$
Capsule geometry is critical in pressure vessel engineering. The hemispherical end caps distribute internal pressure more evenly than flat ends, reducing stress concentrations. This is why propane tanks, oxygen cylinders, and submarine hulls use capsule shapes. In pharmaceuticals, capsule dimensions determine dosage volume and swallowability. In architecture, capsule-shaped buildings and pods are used for their aerodynamic and structural properties. In biology, many bacteria (such as E. coli) are capsule-shaped, and biologists model their volume to calculate cell density and growth rates. In sports, the dimensions of fields with semicircular ends (like athletic tracks) use the same mathematics in two dimensions. This calculator gives you instant, precise results for all capsule dimensions.
Enter the radius $$r$$ and the cylinder height $$h$$ (the straight section length, excluding the rounded ends). The calculator computes the cylinder volume ($$\pi r^2 h$$) and the sphere volume ($$\frac{4}{3}\pi r^3$$) separately, then sums them for the total volume. Surface area is computed as $$2\pi r(2r + h)$$. Total length adds the two hemisphere radii to the cylinder height.
The total volume is the full internal capacity of the capsule. The hemisphere portions volume and cylinder portion volume show how the total is distributed — useful for understanding how much the rounded ends contribute. The surface area is the total outer area, relevant for material estimation, coating, or heat transfer calculations. The total length is the end-to-end measurement. When $$h = 0$$, the capsule degenerates into a sphere.
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Results
A capsule with r = 5 and h = 10 has total volume ~1309 cubic units. The cylinder holds ~785 and the sphere ends hold ~524.
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Results
A Size 0 capsule approximation with r = 3.65 mm and h = 14.3 mm yields ~802 mm³ volume.
The cylinder height $$h$$ is the length of the straight cylindrical section only. The total length is the overall end-to-end dimension: $$L = h + 2r$$. The difference of $$2r$$ accounts for the two hemispherical caps, each extending $$r$$ beyond the cylinder.
When $$h = 0$$, the cylindrical section disappears and the capsule becomes a perfect sphere of radius $$r$$. The volume reduces to $$\frac{4}{3}\pi r^3$$ and the surface area to $$4\pi r^2$$, which are the standard sphere formulas.
Hemispherical end caps distribute internal pressure uniformly across their surface, creating equal stress in all directions (membrane stress). Flat ends would require much thicker material to resist the same pressure. This makes capsule shapes the most material-efficient design for pressurized containers, following the engineering principle that $$\sigma = \frac{pR}{2t}$$ for a hemisphere versus $$\sigma = \frac{pR}{t}$$ for a cylinder wall.
Substitute $$h = L - 2r$$ into the volume formula: $$V = \pi r^2(L - 2r) + \frac{4}{3}\pi r^3 = \pi r^2 L - \frac{2}{3}\pi r^3$$. This cubic equation in $$r$$ can be solved numerically for any given $$V$$ and $$L$$.
Yes. Orientation does not affect volume or surface area. The calculator assumes the capsule is symmetric, so it works regardless of whether it stands upright or lies horizontally. Only the labeling changes — 'height' becomes 'length' conceptually, but the mathematics is identical.
A capsule always has more volume than a cylinder with the same radius and height because of the added hemispherical caps. The additional volume is exactly $$\frac{4}{3}\pi r^3$$ (one full sphere). For very long capsules ($$h \gg r$$), the hemisphere contribution becomes relatively small; for short capsules ($$h \approx 0$$), the sphere dominates.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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