Spherical Cap Volume Calculator
This calculator finds the volume and surface areas of a spherical cap (a portion of a sphere cut off by a plane) based on its dimensions.
Enter any two of the following three dimensions: the Sphere Radius (R), the Cap Base Radius (r), or the perpendicular Cap Height (h). The calculator will determine the missing dimension and calculate the cap's properties. Ensure consistent units.
Enter Spherical Cap Dimensions (Any Two)
Understanding Spherical Cap Volume & Formulas
What is a Spherical Cap?
A spherical cap is a portion of a sphere cut off by a plane. Imagine slicing the top off an orange – the piece you remove is a spherical cap. If the plane passes through the center, it creates a hemisphere (a special case where cap height h equals sphere radius R). It's defined by the radius of the original sphere (R), the radius of the flat circular base created by the cut (r), and the perpendicular height of the cap itself (h).
Spherical Cap Volume Formula
The primary spherical cap volume formula uses the sphere radius (R) and the cap height (h):
V = (1/3) * π * h² * (3R - h)
This formula calculates the volume of dome-like shapes or segments cut from spheres.
Relationship Formulas (R, r, h)
Since R, r, and h are related by the Pythagorean theorem in a cross-section (R² = r² + (R-h)²), if you know any two, you can find the third:
- Given Base Radius (r) and Height (h):
R = (h² + r²) / (2h) - Given Sphere Radius (R) and Height (h):
r = √(2Rh - h²)(Requires 2Rh ≥ h², which is true if h ≤ 2R) - Given Sphere Radius (R) and Base Radius (r): (Assuming h ≤ R, the smaller cap)
h = R - √(R² - r²)(Requires R ≥ r)
This calculator uses these relationships if you only provide two dimensions.
Spherical Cap Surface Area Formulas
- Curved Surface Area (CSA): The area of the curved 'dome' part only. Interestingly, it depends only on R and h.
CSA = 2 * π * R * h - Base Area (BA): The area of the flat circular base.
BA = π * r² - Total Surface Area (TSA): The sum of the curved area and the base area.
TSA = CSA + BA = 2πRh + πr²
Example Calculation (Provided in Original Text)
EX: Jack cuts a cap from James' golf ball. Sphere Radius (R) = 1.68 inches, Cap Height (h) = 0.3 inches. Calculate the volume:
V = (1/3) * π * h² * (3R - h) = (1/3) * π * (0.3)² * (3 * 1.68 - 0.3)
V = (1/3) * π * 0.09 * (5.04 - 0.3) = (1/3) * π * 0.09 * 4.74
Result: V ≈ 0.447 cubic inches (in³).
Real-Life Spherical Cap Examples
Click on an example to see the dimensions and results:
Example 1: Dome Roof Volume (R=20m, h=5m)
Scenario: Estimate the volume inside a dome roof, modeled as a spherical cap.
Known Values: Sphere Radius (R) = 20 m, Cap Height (h) = 5 m.
Calculation Results:
R = 20 m
r ≈ 13.23 m
h = 5 m
Volume ≈ 1440.00 m³
Curved Surface Area ≈ 628.32 m²
Base Area ≈ 547.44 m²
Total Surface Area ≈ 1175.76 m²
Example 2: Contact Lens Volume (r=6mm, h=1mm)
Scenario: Model a contact lens as a thin spherical cap to estimate its material volume (simplified).
Known Values: Cap Base Radius (r) = 6 mm, Cap Height (h) = 1 mm.
Calculation Results:
R = 18.5 mm
r = 6 mm
h = 1 mm
Volume ≈ 57.09 mm³
Curved Surface Area ≈ 116.24 mm²
Base Area ≈ 113.10 mm²
Total Surface Area ≈ 229.34 mm²
Example 3: Liquid in a Spherical Bowl (R=10cm, h=4cm)
Scenario: A hemispherical bowl (R=10cm) is filled with water to a depth (height) of 4cm. Find the volume of water.
Known Values: Sphere Radius (R) = 10 cm, Cap Height (h) = 4 cm.
Calculation Results:
R = 10 cm
r = 8 cm
h = 4 cm
Volume ≈ 435.63 cm³
Curved Surface Area ≈ 251.33 cm²
Base Area ≈ 201.06 cm²
Total Surface Area ≈ 452.39 cm²
Example 4: Planetary Ice Cap Volume (R=3000km, r=1000km)
Scenario: Estimate the volume of a planet's polar ice cap, modeled as a spherical cap.
Known Values: Planet Radius (R) = 3000 km, Ice Cap Base Radius (r) = 1000 km.
Calculation Results:
R = 3000 km
r = 1000 km
h ≈ 171.74 km
Volume ≈ 272,407,086.50 km³
Curved Surface Area ≈ 3,237,581.37 km²
Base Area ≈ 3,141,592.65 km²
Total Surface Area ≈ 6,379,174.02 km²
Example 5: Cut Section of a Ball (R=5in, h=2in)
Scenario: A solid ball is cut, removing a spherical cap section.
Known Values: Ball/Sphere Radius (R) = 5 inches, Height of cut piece (h) = 2 inches.
Calculation Results:
R = 5 in
r = 4 in
h = 2 in
Volume ≈ 54.45 in³
Curved Surface Area ≈ 62.83 in²
Base Area ≈ 50.27 in²
Total Surface Area ≈ 113.10 in²
Example 6: Liquid in Spherical Tank (R=2m, h=0.5m)
Scenario: A spherical tank is partially filled with liquid.
Known Values: Tank/Sphere Radius (R) = 2 meters, Liquid Depth/Height (h) = 0.5 meters.
Calculation Results:
R = 2 m
r ≈ 1.323 m
h = 0.5 m
Volume ≈ 1.44 m³
Curved Surface Area ≈ 6.28 m²
Base Area ≈ 5.50 m²
Total Surface Area ≈ 11.78 m²
Example 7: Architectural Dome Feature (r=1m, h=0.3m)
Scenario: A small decorative dome is part of a larger design.
Known Values: Radius of the dome's base (r) = 1 meter, Height of the dome (h) = 0.3 meters.
Calculation Results:
R ≈ 1.817 m
r = 1 m
h = 0.3 m
Volume ≈ 0.49 m³
Curved Surface Area ≈ 3.43 m²
Base Area ≈ 3.14 m²
Total Surface Area ≈ 6.57 m²
Example 8: Optical Lens Surface (R=50mm, r=10mm)
Scenario: Model the curved part of a simple optical lens as a spherical cap.
Known Values: Sphere Radius of curvature (R) = 50 mm, Lens Base Radius (r) = 10 mm.
Calculation Results:
R = 50 mm
r = 10 mm
h ≈ 1.01 mm
Volume ≈ 159.72 mm³
Curved Surface Area ≈ 317.31 mm²
Base Area ≈ 314.16 mm²
Total Surface Area ≈ 631.47 mm²
Example 9: Machined Spherical Indentation (R=2in, h=0.5in)
Scenario: Material removed when creating a spherical cap indentation during machining.
Known Values: Original Sphere Radius (R) = 2 inches, Depth of cut / Cap Height (h) = 0.5 inches.
Calculation Results:
R = 2 in
r ≈ 1.323 in
h = 0.5 in
Volume ≈ 1.44 in³
Curved Surface Area ≈ 6.28 in²
Base Area ≈ 5.48 in²
Total Surface Area ≈ 11.76 in²
Example 10: Hemisphere Volume (R=5 units)
Scenario: Calculate the volume of a hemisphere (half a sphere). For a hemisphere, h=R and r=R.
Known Values: Sphere Radius (R) = 5 units, Cap Height (h) = 5 units (since h=R for a hemisphere).
Note: You can also input R=5 and Cap Base Radius (r)=5, and the calculator will correctly determine h=5.
Calculation Results:
R = 5 units
r = 5 units
h = 5 units
Volume ≈ 261.80 units³
Curved Surface Area ≈ 157.08 units²
Base Area ≈ 78.54 units²
Total Surface Area ≈ 235.62 units²
Understanding Volume Measurement
Volume is the quantification of the three-dimensional space... (Content about volume units, definitions, etc. can go here following your template structure. I've left placeholder text.)
Common Volume Units Reference
Ensure your input dimensions (R, r, h) use a consistent unit... (Units table placeholder following your template structure.)
| Linear Unit | Corresponding Volume Unit |
|---|---|
| Millimeters (mm) | Cubic Millimeters (mm³) |
| Centimeters (cm) | Cubic Centimeters (cm³) |
| Inches (in) | Cubic Inches (in³) |
| Feet (ft) | Cubic Feet (ft³) |
| Meters (m) | Cubic Meters (m³) |
| Kilometers (km) | Cubic Kilometers (km³) |
Frequently Asked Questions about Spherical Cap Volume
1. What is the main spherical cap volume formula?
The most common formula uses the sphere radius (R) and cap height (h): V = (1/3) * π * h² * (3R - h).
2. How does this calculator work if I only provide two dimensions?
It uses the geometric relationship between R, r, and h (R² = r² + (R-h)²) to calculate the missing third dimension first. Then it uses the primary volume formula V = (1/3)πh²(3R - h) and the area formulas.
3. What's the difference between a spherical cap and a sphere segment?
A spherical cap is cut by one plane. A spherical segment is the portion of a sphere between *two* parallel cutting planes.
4. What is a hemisphere?
A hemisphere is a special spherical cap where the cutting plane goes through the center of the sphere. In this case, the cap height (h) is equal to the sphere radius (R), and the base radius (r) is also equal to R.
5. How is the Curved Surface Area (CSA) of the cap calculated?
The area of just the curved part is CSA = 2 * π * R * h. Notice it doesn't depend on the base radius 'r'.
6. What are the limitations on the input values?
- All values (R, r, h) must be valid, non-negative numbers.
- You must provide exactly two dimensions.
- The provided or calculated dimensions must form a valid spherical cap geometry (e.g., h cannot be greater than 2R, r cannot be greater than R, R must be positive).
7. What units should I use?
Use consistent linear units (like cm, meters, inches, feet) for all inputs. The calculator performs the calculation; the units in the results will correspond to the units you input (e.g., if you input meters, volume is in m³, area in m²).
8. Is this the same as the volume of a dome?
Many architectural domes are shaped like spherical caps, so this formula and calculator can often be used to approximate their volume or surface area, assuming the dome is a perfect spherical cap.
9. Can I calculate the volume of the *remaining* part of the sphere after the cap is cut?
Yes. First, calculate the volume of the full sphere (Vsphere = 4/3 * π * R³). You can use this calculator by inputting R and any other dimension to find R, then calculate the full sphere volume separately. Calculate the volume of the cap (Vcap) using this tool. The remaining volume is Vremaining = Vsphere - Vcap.
10. What if I know the base radius (r) and sphere radius (R)? How does the calculator find 'h'?
It uses the formula h = R - √(R² - r²). This calculates the height of the *smaller* cap cut from the sphere (the one where the height h is less than or equal to R).
