On This Page
- Capsule volume formula
- How to calculate capsule volume
- Total length vs. cylinder height
- Capsule tank volume
- Worked examples
- Capsule volume units, liters, and gallons
- Frequently asked questions
Capsule Volume Formula
A capsule is a cylinder plus two hemispheres. Since two hemispheres of the same radius equal one complete sphere, the total volume is cylinder volume plus sphere volume.
- V = capsule volume
- r = radius of the cylinder and rounded ends (if diameter is given: r = d ÷ 2)
- h = cylinder height, the straight middle section only, not the full end-to-end length
The first term (πr²h) is the middle cylinder; the second term ((4/3)πr³) is the two rounded ends combined. If the problem gives total capsule length instead of cylinder height, use h = total length − 2r before substituting.
How to Calculate Capsule Volume
Split the shape into two familiar parts, a cylinder and a sphere, and add the results. Both parts use the same radius.
- Find radius r (divide diameter by 2 if needed)
- Find cylinder height h (subtract 2r from total length if needed)
- Cylinder volume: πr²h
- Rounded ends: (4/3)πr³
- Total: add both volumes and write in cubic units
Total Length vs. Cylinder Height
Capsule problems often give total length instead of cylinder height. Total length includes the straight cylinder section plus two radii at the rounded ends.
For example, if total length is 24 cm and radius is 4 cm, then cylinder height is 24 - 8 = 16 cm.
Capsule Tank Volume
Horizontal and vertical capsule tanks, including propane tanks, above-ground fuel storage, and industrial pressure vessels, use the same full-volume formula regardless of orientation. Enter the radius and straight cylinder section length to get the full tank capacity in liters or gallons.
Orientation only matters for partial-fill calculations. When a horizontal capsule tank is not full, the liquid surface cuts through the rounded ends at an angle, so a simple depth-based fraction gives the wrong answer. This calculator gives full tank capacity; for partial fill of horizontal tanks, specialized engineering tools handle the cross-section geometry.
For propane tanks and similar vessels, the manufacturer label states rated water capacity, the actual usable capacity for the stored product is usually less. Use the inside radius and inside straight-section length for the most accurate capacity estimate. If the vessel is a smooth elongated shape without a straight cylindrical middle section, such as an egg-shaped digester tank, the Ellipsoid Volume Calculator is the better fit.
Worked Examples
In capsule problems, the key is separating the straight middle from the rounded ends. Watch whether the problem gives radius, diameter, cylinder height, or total length.
Example 1: Capsule Volume From Radius and Cylinder Height
Problem: A capsule-shaped solid has radius 3 cm and cylinder height 10 cm. Find the total volume. Use 3.14159 for π.
- Radius = 3 cm, cylinder height = 10 cm
- Cylinder volume = π × 3² × 10 = 282.74 cm³
- Rounded-end volume = (4/3)π × 3³ = 113.10 cm³
- Total volume = 282.74 + 113.10 = 395.84 cm³
Answer: The capsule volume is about 395.8 cm³.
Example 2: Capsule Volume From Total Length
Problem: A pill-shaped container is 24 cm long from end to end and has radius 4 cm. Find its volume.
- Cylinder height = 24 - (2 × 4) = 16 cm
- Cylinder volume = π × 4² × 16 = 804.25 cm³
- Rounded-end volume = (4/3)π × 4³ = 268.08 cm³
- Total volume = 1,072.33 cm³
Answer: The capsule volume is about 1,072.3 cm³.
Example 3: Horizontal Capsule Tank Capacity in Liters
Problem: A horizontal capsule tank has radius 0.6 m and straight cylinder length 2.8 m. Estimate its full capacity in liters.
- Cylinder volume = π × 0.6² × 2.8 = 3.167 m³
- Rounded-end volume = (4/3)π × 0.6³ = 0.905 m³
- Total volume = 3.167 + 0.905 = 4.072 m³
- Liters = 4.072 × 1,000 = 4,072 L
Answer: The tank holds about 4,072 liters when full.
Example 4: Diameter Given Instead of Radius
Problem: A capsule has diameter 8 inches and cylinder height 14 inches. Find the volume.
- Radius = 8 ÷ 2 = 4 in
- Cylinder volume = π × 4² × 14 = 703.72 in³
- Rounded-end volume = (4/3)π × 4³ = 268.08 in³
- Total volume = 971.80 in³
Answer: The capsule volume is about 971.8 cubic inches.
Example 5: Vertical Capsule Tank Full Capacity
Problem: A vertical capsule tank has radius 1.2 ft and cylinder height 5 ft. Estimate the full volume in cubic feet.
- Cylinder volume = π × 1.2² × 5 = 22.62 ft³
- Rounded-end volume = (4/3)π × 1.2³ = 7.24 ft³
- Total volume = 22.62 + 7.24 = 29.86 ft³
Answer: The vertical capsule tank holds about 29.9 ft³ when full.
Capsule Volume Units, Liters, and Gallons
Capsule volume is written in cubic units. For tanks and containers, cubic volume is often converted to liters or gallons.
| Input Unit | Volume Unit | Useful Conversion |
|---|---|---|
| Centimeters | cm³ | 1,000 cm³ = 1 liter |
| Meters | m³ | 1 m³ = 1,000 liters |
| Inches | in³ | 231 in³ = 1 US gallon |
| Feet | ft³ | 1 ft³ ≈ 7.48052 US gallons |
Frequently Asked Questions
How do you calculate capsule volume?
Use V = πr²h + (4/3)πr³. Calculate the middle cylinder (πr²h) and both rounded ends together as one sphere ((4/3)πr³), then add. Both terms use the same radius. If diameter is given, divide by 2 first: r = d ÷ 2.
How do you calculate capsule volume from total length?
Find cylinder height first: h = total length − 2r. For a capsule 24 cm long with radius 4 cm: cylinder height = 24 − 8 = 16 cm. Then apply the full formula.
Is horizontal capsule tank volume different from vertical?
Full capacity is identical regardless of orientation. Orientation only affects partial-fill calculations, when a horizontal tank is not full, the liquid surface cuts through the rounded ends at an angle, making depth-based estimates inaccurate.
What everyday objects are capsule-shaped?
Pharmaceutical capsule tablets are the clearest example. Propane tanks, industrial pressure vessels, and railway tank cars use the shape because hemispheres distribute internal pressure stress evenly with no sharp stress-concentration edges. Gel capsule supplements also approximate it.
Why does the formula use a full sphere rather than two separate hemispheres?
Two hemispheres of the same radius equal exactly one sphere, so the calculation simplifies to the sphere volume formula (4/3)πr³ for both ends combined. Calculating each hemisphere as (2/3)πr³ and doubling gives the same result, it is just one extra step with no benefit.
References
- Spherical Cylinder, Wolfram MathWorld: mathematical definition of the capsule shape, volume formula, and surface area derivation.
- Cylinder, Wolfram MathWorld: cylinder volume formula (πr²h) and surface area, the basis for the straight-section term.
- Capsule, Math is Fun: practical explanation of total length vs. cylinder height with worked examples.