Capsule Segment Calculator

Name: Capsule Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at a capsule segment. This is a capsule, which is divided in its length and closed at the slice plane. The division does not have to be centered, but it must be parallel to a centered division, at any height. The capsule segment consists of a bisected spherical cap, with a fitting cylindrical segment inserted between.
Enter radius of the sphere and height of the cylinder, as well as the filling height of the capsule. Choose the number of decimal places, then click Calculate.

Formulas:

s = 2 \cdot \sqrt{2 \cdot r \cdot f - f^{2}}

l = h + \sqrt{s^{2} - f^{2}}

A = 2 h r \cdot \arccos (1 - \frac{f}{r}) + π \cdot [2 r f + f \cdot (2 r - f)] + 2 h \cdot \sqrt{2 r f - f^{2}}

V = h \cdot [r^{2} \cdot \arccos (1 - \frac{f}{r}) - \frac{\sqrt{2 r f - f^{2}} \cdot (r - f)}{2}] + \frac{f^{2} \cdot π \cdot (3 r - f)}{3}

pi:

π = 3.141592653589793 ...

Radius, length and heights have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1. The s in the formulas is the filling width of the cylindrical segment.

Capsules of this shape serve as containers for various items, such as medications. When not completely filled and lying horizontally, the contents take the form of a capsule segment. This shape can therefore also be referred to as a capsule filling.

A capsule segment has two surfaces. A flat surface shaped like a stadium, and a curved surface composed of the two halves of the calotte of the spherical cap and the lateral surface of the cylindrical segment. A continuous edge lies between them. The vertical slice plane through a capsule segment is a stadium segment.
The formulas for calculating surface area and volume are quite complex because these values do not increase linearly with the fill level. This is due to the curvature of the cylindrical segment. The relationship can be expressed using the cosine function, resulting in an arccos term in the formulas, where a π would appear for the original circle of the cylinder base. If the division of the capsule is not exactly in the middle, the original sphere of the capsule is no longer present in the capsule segment with its full radius.

Calculation of volume and other dimensions: capsule segment

Last updated on 07/19/2026. Author: Jürgen Kummer

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