Capsule Volume Calculator
This calculator finds the volume, total surface area, and total length of a capsule shape, which consists of a cylinder with two hemispherical ends.
Enter the radius (r) common to the cylinder and hemispheres, and the height (h) of the cylindrical portion only to calculate the capsule's properties. Ensure consistent units.
Enter Capsule Dimensions
Understanding Capsule Volume & Formulas
What is a Capsule?
A capsule is a three-dimensional geometric shape composed of a cylinder connected to two hemispherical ends (a hemisphere is half a sphere). The radius (r) is the same for the cylinder and the hemispheres. The height (h) in the formulas refers specifically to the height of the cylindrical portion only.
The Capsule Volume Formula
The capsule volume formula is the sum of the volume of the central cylinder and the volume of the two hemispherical ends (which together make one full sphere):
V = Vcylinder + Vsphere = (π * r² * h) + ((4/3) * π * r³)
This can be simplified to:
V = π * r² * (h + (4/3) * r)
Where:
- V is the Total Volume
- π (Pi) is approximately 3.14159...
- r is the radius
- h is the height of the cylindrical portion
Other Capsule Formulas
- Total Surface Area (SA): The sum of the lateral surface area of the cylinder and the surface area of the two hemispherical ends (which equals the surface area of one full sphere).
SA = SAcylinder_side + SAsphere = (2 * π * r * h) + (4 * π * r²)
Simplified:SA = 2 * π * r * (h + 2 * r) - Total Length (L): The overall length of the capsule from tip to tip.
L = h + r + r = h + 2r
Example Calculation (Provided in Original Text)
EX: A time capsule has a radius (r) of 1.5 ft and a cylindrical portion height (h) of 3 ft. Calculate the volume:
V = π * r² * (h + (4/3) * r) = π * (1.5)² * (3 + (4/3) * 1.5) = π * 2.25 * (3 + 2) = π * 2.25 * 5 = 11.25π
Result: V ≈ 35.34 cubic feet (ft³).
Real-Life Capsule Volume Examples
Click on an example to see the step-by-step calculation:
Example 1: Gelatin Capsule (#0 Size)
Scenario: Estimate the internal volume of a standard #0 size gelatin capsule.
1. Known Values: Approx. Radius (r) ≈ 0.15 inches. Total Length ≈ 0.85 inches.
2. Find Cyl. Height (h): h = Total Length - 2r = 0.85 - 2 * 0.15 = 0.85 - 0.30 = 0.55 inches.
3. Formula: V = π * r² * (h + (4/3) * r)
4. Calculation: V ≈ π * (0.15)² * (0.55 + (4/3) * 0.15) ≈ π * 0.0225 * (0.55 + 0.20) = π * 0.0225 * 0.75
5. Result: V ≈ 0.053 cubic inches (about 0.87 mL).
Conclusion: A #0 gel cap holds roughly 0.05 cubic inches.
Example 2: Propane Tank (20 lb Cylinder Part)
Scenario: Calculate the approximate volume of a typical 20 lb propane tank (treating ends as hemispheres).
1. Known Values: Approx. Radius (r) ≈ 6 inches, Cylindrical Height (h) ≈ 12 inches.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V ≈ π * (6)² * (12 + (4/3) * 6) = π * 36 * (12 + 8) = π * 36 * 20 = 720π
4. Result: V ≈ 2261.9 cubic inches (about 9.8 US gallons).
Conclusion: The propane tank has an approximate internal volume of 2260 cubic inches.
Example 3: Capsule-Shaped Storage Tank
Scenario: Find the volume of a capsule-shaped industrial storage tank.
1. Known Values: Radius (r) = 1 meter, Cylindrical Height (h) = 3 meters.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (1)² * (3 + (4/3) * 1) ≈ π * 1 * (3 + 1.333) = 4.333π
4. Result: V ≈ 13.61 cubic meters.
Conclusion: The tank holds about 13.6 cubic meters.
Example 4: Boiler Tank Volume
Scenario: Calculate the volume of a small cylindrical boiler with hemispherical ends.
1. Known Values: Radius (r) = 1 foot, Cylindrical Height (h) = 4 feet.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (1)² * (4 + (4/3) * 1) ≈ π * 1 * (4 + 1.333) = 5.333π
4. Result: V ≈ 16.76 cubic feet.
Conclusion: The boiler tank volume is approximately 16.8 cubic feet.
Example 5: Thermos Inner Flask Volume
Scenario: Estimate the volume of the inner vacuum flask of a thermos, assuming capsule shape.
1. Known Values: Radius (r) = 3 cm, Cylindrical Height (h) = 20 cm.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (3)² * (20 + (4/3) * 3) = π * 9 * (20 + 4) = π * 9 * 24 = 216π
4. Result: V ≈ 678.6 cubic cm (or 678.6 mL).
Conclusion: The thermos flask holds roughly 680 mL.
Example 6: Small Buoy Volume
Scenario: Calculate the volume of a small capsule-shaped marker buoy.
1. Known Values: Radius (r) = 0.2 meters, Cylindrical Height (h) = 0.5 meters.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (0.2)² * (0.5 + (4/3) * 0.2) ≈ π * 0.04 * (0.5 + 0.267) = π * 0.04 * 0.767
4. Result: V ≈ 0.096 cubic meters (or 96 Liters).
Conclusion: The buoy has a volume of about 0.1 cubic meters.
Example 7: CO2 Cartridge Volume (12g Size Approx)
Scenario: Estimate the internal volume of a 12g CO2 cartridge.
1. Known Values: Approx. Radius (r) ≈ 0.9 cm, Approx. Cylindrical Height (h) ≈ 6 cm.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (0.9)² * (6 + (4/3) * 0.9) ≈ π * 0.81 * (6 + 1.2) = π * 0.81 * 7.2
4. Result: V ≈ 18.32 cubic cm (or 18.3 mL).
Conclusion: A 12g CO2 cartridge has an internal volume around 18 cubic centimeters.
Example 8: Generic Pill Container Volume
Scenario: Find the volume of a small plastic pill container shaped like a capsule.
1. Known Values: Radius (r) = 1 cm, Cylindrical Height (h) = 4 cm.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (1)² * (4 + (4/3) * 1) ≈ π * 1 * (4 + 1.333) = 5.333π
4. Result: V ≈ 16.76 cubic cm (or 16.8 mL).
Conclusion: The container holds about 17 mL.
Example 9: Large Industrial Tank Volume
Scenario: Calculate the volume of a large industrial capsule tank.
1. Known Values: Radius (r) = 2 meters, Cylindrical Height (h) = 10 meters.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (2)² * (10 + (4/3) * 2) ≈ π * 4 * (10 + 2.667) = π * 4 * 12.667
4. Result: V ≈ 159.17 cubic meters.
Conclusion: The large tank has a volume of approximately 159 cubic meters.
Example 10: Hot Water Heater Tank Volume
Scenario: Estimate the volume of a typical residential hot water heater tank (assuming capsule shape).
1. Known Values: Approx. Radius (r) = 10 inches, Approx. Cylindrical Height (h) = 40 inches.
2. Formula: V = π * r² * (h + (4/3) * r)
3. Calculation: V = π * (10)² * (40 + (4/3) * 10) ≈ π * 100 * (40 + 13.333) = π * 100 * 53.333
4. Result: V ≈ 16755 cubic inches (about 72.5 US gallons).
Conclusion: The hot water tank holds roughly 16,750 cubic inches.
Understanding Volume Measurement
Volume is the quantification of the three-dimensional space...
Common Volume Units Reference
Ensure your input radius and cylinder height use a consistent unit...
Frequently Asked Questions about Capsule Volume
1. What is the capsule volume formula?
The volume (V) combines a cylinder and a sphere: V = (Volume of Cylinder) + (Volume of Sphere) = (π * r² * h) + ((4/3) * π * r³). A simplified version is V = π * r² * (h + (4/3) * r).
2. What do 'r' and 'h' represent in the formula?
'r' is the radius of both the cylindrical part and the hemispherical ends. 'h' is the height (or length) of *only* the straight, cylindrical middle section.
3. How do I find the cylindrical height (h) if I know the total length (L)?
The total length (L) includes the cylindrical height plus the radius of each hemispherical end (L = h + r + r = h + 2r). So, if you know L and r, you can find h: h = L - 2r.
4. What is the formula for the Total Surface Area of a capsule?
It's the surface area of the cylindrical side plus the surface area of the two hemispherical ends (which equals one full sphere's surface area): SA = (2 * π * r * h) + (4 * π * r²), or simplified: SA = 2 * π * r * (h + 2r).
5. What is the Total Length of the capsule?
The total length (L) from tip to tip is the height of the cylindrical part (h) plus the radius (r) from each of the two hemispherical ends: L = h + 2r.
6. Are medication capsules always this exact shape?
This formula calculates the volume of a perfect geometric capsule. Real medication capsules (like gel caps) are very close to this shape, but manufacturing might cause slight variations.
7. What units should I use for radius and height?
Use any consistent unit of length (e.g., mm, cm, inches, meters). The resulting Volume will be in the corresponding cubic units (mm³, cm³, in³, m³), Surface Area in square units, and Total Length in linear units.
8. Can the cylindrical height (h) be zero?
Yes. If h = 0, the formula simplifies to V = (4/3) * π * r³, which is just the volume of a sphere. A capsule with zero cylinder height is simply a sphere.
9. How does this relate to the "volume of a circle"?
The capsule formula uses the radius 'r', which defines the circular cross-section of the cylinder and the base of the hemispheres. The volume calculation itself relies on the formulas for cylinder and sphere volumes, both derived from circular properties.
10. Is this the same as an ellipsoid?
No. An ellipsoid is like a stretched or squashed sphere and has three different semi-axes (a, b, c). A capsule has a distinct cylindrical middle section and requires only radius (r) and cylinder height (h).
11. Where might I encounter capsule shapes?
Common examples include medication capsules, propane tanks, storage silos, pressure vessels, thermos interiors, buoys, and some types of packaging.
