Capsule Calculator
Calculate capsule (spherocylinder) properties including volume, surface area, and dimensions. Includes pharmaceutical pill size references.
V = (4/3)pir^3 + pir^2aCapsule Properties
Volume
395.8407 cm^3
3D Visualization
Capsule Input
Radius of the hemispherical ends and cylinder
Length of the cylindrical middle section
Real-World Capsule Examples
Quick-start with common scenarios
All Capsule Properties
Total Volume
395.8407 cm^3
V = (4/3)pir^3 + pir^2a
Hemisphere Volume
113.0973 cm^3
= (4/3)pir^3
Cylinder Volume
282.7433 cm^3
= pir^2a
Total Surface Area
301.5929 cm^2
SA = 4pir^2 + 2pira
Hemisphere Surface
113.0973 cm^2
= 4pir^2
Cylinder Surface
188.4956 cm^2
= 2pira
Exact Values (in terms of pi)
Volume
126pi cm^3
Surface Area
96pi cm^2
2D Stadium Cross-Section
The 2D cross-section of a capsule is called a stadium (or discorectangle).
Stadium Area
88.2743 cm^2
= pir^2 + 2ra
Stadium Perimeter
38.8496 cm
= 2pir + 2a
Capsule Formulas
Volume Formulas
- V = V_sphere + V_cylinder
- V = (4/3)pir^3 + pir^2a
- V = pir^2(4r/3 + a)
- From total length: V = pir^2(L - 2r/3)
Surface Area Formulas
- SA = SA_sphere + SA_lateral
- SA = 4pir^2 + 2pira
- SA = 2pir(2r + a)
- From total length: SA = 2pir x L
Derivation: A capsule is a cylinder of height a, capped with two hemispheres of radius r. The two hemispheres together form one complete sphere. There are no circular bases to add since the hemispheres connect directly to the cylinder.
Capsule Properties
Volume
395.8407 cm^3
?How to Calculate Capsule Properties
Capsule formulas: A capsule is a cylinder capped with two hemispheres. Volume = (4/3)pir^3 + pir^2a (sphere volume + cylinder volume), where r is radius and a is cylinder length. Surface Area = 4pir^2 + 2pira (sphere surface + cylinder lateral). Total length L = 2r + a.
What is a Capsule?
A capsule (also called spherocylinder or stadium of revolution) is a three-dimensional shape consisting of a cylinder with hemispherical caps on both ends. It can be thought of as a cylinder smoothly rounded at both ends. The shape is named after pharmaceutical capsules which commonly use this form. The 2D equivalent is called a stadium or discorectangle.
Key Facts About Capsules
- Volume = (4/3)pir^3 + pir^2a (hemisphere pair + cylinder)
- Surface Area = 4pir^2 + 2pira (sphere + cylinder lateral)
- Total length = 2r + a (two radii plus cylinder length)
- A capsule is also called a spherocylinder or stadium of revolution
- The 2D cross-section is called a stadium or discorectangle
- Pharmaceutical capsules come in standard sizes 000 through 5
- Capsule shapes are common in chemistry for molecular models
- Stadium shapes are used in running track design
Quick Answer
Capsule formulas: A capsule is a cylinder capped with two hemispheres. Volume = (4/3)pir^3 + pir^2a (sphere volume + cylinder volume), where r is radius and a is cylinder length. Surface Area = 4pir^2 + 2pira (sphere surface + cylinder lateral). Total length L = 2r + a.
Practice Capsule Problems
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Practice with 4 problems to test your understanding.
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Frequently Asked Questions
Capsule volume is the sum of a sphere and a cylinder: V = (4/3)pir^3 + pir^2a, where r is the radius and a is the cylinder length (not total length). The first term is the volume of two hemispheres (= one sphere), and the second term is the cylinder volume.
Surface Area = 4pir^2 + 2pira. The 4pir^2 is the surface area of a complete sphere (two hemispheres joined), and 2pira is the lateral surface area of the cylinder (no bases since they connect to the hemispheres).
Side length (a) is just the cylindrical portion in the middle. Total length (L) includes the two hemispherical ends: L = 2r + a. When you measure a physical capsule end-to-end, that is the total length.
A stadium (also called discorectangle or obround) is the 2D cross-section of a capsule. It looks like a rectangle with semicircles attached to opposite ends. Stadium Area = pir^2 + 2ra. Stadium Perimeter = 2pir + 2a.
Pharmaceutical capsules range from size 000 (largest, ~26mm long) to size 5 (smallest, ~11mm long). Common sizes: 00 (23.3mm), 0 (21.7mm), 1 (19.4mm), 2 (18.0mm). Volume capacity ranges from 1.37mL (size 000) to 0.13mL (size 5).
Spherocylinder is another name for capsule - a cylinder with hemispherical caps. It is used in physics and chemistry to model molecular shapes and in computer graphics for collision detection.
If you know total length L and radius r, first find side length: a = L - 2r. Then use V = (4/3)pir^3 + pir^2a. Or equivalently: V = pir^2(L - r + r/3) = pir^2(L - 2r/3).
Capsule shapes are easy to swallow due to their smooth, elongated form. They also provide good volume-to-surface-area ratio for drug containment, are structurally strong, and can be easily manufactured by joining two half-shells.
Last updated: 2025-01-15
Capsule Properties
Volume
395.8407 cm^3