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 step-by-step calculation (dimensions are often approximate):
Example 1: Dome Roof Volume
Scenario: Estimate the volume inside a dome roof, modeled as a spherical cap.
1. Known Values: Full Sphere Radius if completed (R) = 20 m, Height of the dome cap (h) = 5 m.
2. Calculate Missing Dim (r): r = √(2Rh - h²) = √(2*20*5 - 5²) = √(200 - 25) = √175 ≈ 13.229 m.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (5)² * (3*20 - 5) = (1/3) * π * 25 * (60 - 5) = (1/3) * π * 25 * 55
5. Result: V ≈ 1439.896 cubic meters.
Conclusion: The dome encloses about 1440 cubic meters.
Example 2: Contact Lens Volume (Approx)
Scenario: Model a contact lens as a thin spherical cap to estimate its material volume (simplified).
1. Known Values: Base Radius (r) ≈ 6 mm, Height (h) ≈ 1 mm.
2. Calculate Missing Dim (R): R = (h² + r²) / (2h) = (1² + 6²) / (2*1) = (1 + 36) / 2 = 37 / 2 = 18.5 mm.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (1)² * (3*18.5 - 1) = (1/3) * π * 1 * (55.5 - 1) = (1/3) * π * 54.5
5. Result: V ≈ 57.094 cubic mm.
Conclusion: The simplified contact lens model has a volume of about 57 cubic millimeters.
Example 3: Liquid in a Spherical Bowl
Scenario: A hemispherical bowl (R=10cm) is filled with water to a depth (height) of 4cm. Find the volume of water.
1. Known Values: Sphere Radius (R) = 10 cm, Cap Height (h) = 4 cm.
2. Calculate Missing Dim (r): r = √(2Rh - h²) = √(2*10*4 - 4²) = √(80 - 16) = √64 = 8 cm.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V = (1/3) * π * (4)² * (3*10 - 4) = (1/3) * π * 16 * (30 - 4) = (1/3) * π * 16 * 26
5. Result: V ≈ 435.633 cubic cm (or 435.633 mL).
Conclusion: There are about 436 mL of water in the bowl.
Example 4: Planetary Ice Cap Volume (Approx)
Scenario: Estimate the volume of a planet's polar ice cap, modeled as a spherical cap.
1. Known Values: Planet Radius (R) ≈ 3000 km, Ice Cap Base Radius (r) ≈ 1000 km.
2. Calculate Missing Dim (h): h = R - √(R² - r²) = 3000 - √(3000² - 1000²) = 3000 - √(9000000 - 1000000) = 3000 - √8000000 ≈ 3000 - 2828.427 = 171.573 km.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (171.573)² * (3*3000 - 171.573) = (1/3) * π * 29437.3 * (9000 - 171.573) = (1/3) * π * 29437.3 * 8828.427
5. Result: V ≈ 271,731,583 cubic kilometers (approx 2.72 × 10⁸ km³).
Conclusion: The model ice cap has a huge volume, roughly 272 million cubic kilometers.
Example 5: Cut Section of a Ball
Scenario: A solid ball is cut, removing a spherical cap section.
1. Known Values: Ball/Sphere Radius (R) = 5 inches, Height of cut piece (h) = 2 inches.
2. Calculate Missing Dim (r): r = √(2Rh - h²) = √(2*5*2 - 2²) = √(20 - 4) = √16 = 4 inches.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V = (1/3) * π * (2)² * (3*5 - 2) = (1/3) * π * 4 * (15 - 2) = (1/3) * π * 4 * 13
5. Result: V ≈ 54.454 cubic inches.
Conclusion: The volume of the removed cap is about 54.5 cubic inches.
Example 6: Liquid in Spherical Tank
Scenario: A spherical tank is partially filled with liquid.
1. Known Values: Tank/Sphere Radius (R) = 2 meters, Liquid Depth/Height (h) = 0.5 meters.
2. Calculate Missing Dim (r): r = √(2Rh - h²) = √(2*2*0.5 - 0.5²) = √(2 - 0.25) = √1.75 ≈ 1.323 meters.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V = (1/3) * π * (0.5)² * (3*2 - 0.5) = (1/3) * π * 0.25 * (6 - 0.5) = (1/3) * π * 0.25 * 5.5
5. Result: V ≈ 1.4399 cubic meters.
Conclusion: The volume of liquid in the tank is about 1.44 cubic meters.
Example 7: Architectural Dome Feature
Scenario: A small decorative dome is part of a larger design.
1. Known Values: Radius of the dome's base (r) = 1 meter, Height of the dome (h) = 0.3 meters.
2. Calculate Missing Dim (R): R = (h² + r²) / (2h) = (0.3² + 1²) / (2*0.3) = (0.09 + 1) / 0.6 = 1.09 / 0.6 ≈ 1.81667 meters.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (0.3)² * (3*1.81667 - 0.3) = (1/3) * π * 0.09 * (5.45 - 0.3) = (1/3) * π * 0.09 * 5.15
5. Result: V ≈ 0.4850 cubic meters.
Conclusion: The small dome feature has a volume of approximately 0.49 cubic meters.
Example 8: Optical Lens Surface (Approx)
Scenario: Model the curved part of a simple optical lens as a spherical cap.
1. Known Values: Lens Base Radius (r) = 10 mm, Sphere Radius of curvature (R) = 50 mm.
2. Calculate Missing Dim (h): h = R - √(R² - r²) = 50 - √(50² - 10²) = 50 - √(2500 - 100) = 50 - √2400 ≈ 50 - 48.9897 = 1.0103 mm.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (1.0103)² * (3*50 - 1.0103) = (1/3) * π * 1.0207 * (150 - 1.0103) = (1/3) * π * 1.0207 * 148.9897
5. Result: V ≈ 159.710 cubic mm.
Conclusion: The approximate volume of the lens cap is about 160 cubic millimeters.
Example 9: Machined Spherical Indentation
Scenario: Material removed when creating a spherical cap indentation during machining.
1. Known Values: Original Sphere Radius (R) = 2 inches, Depth of cut / Cap Height (h) = 0.5 inches.
2. Calculate Missing Dim (r): r = √(2Rh - h²) = √(2*2*0.5 - 0.5²) = √(2 - 0.25) = √1.75 ≈ 1.3228 inches.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V = (1/3) * π * (0.5)² * (3*2 - 0.5) = (1/3) * π * 0.25 * (6 - 0.5) = (1/3) * π * 0.25 * 5.5
5. Result: V ≈ 1.4399 cubic inches.
Conclusion: About 1.44 cubic inches of material was removed.
Example 10: Hemisphere Volume (Special Case)
Scenario: Calculate the volume of a hemisphere (half a sphere).
1. Known Values: Sphere Radius (R) = 5 units. For a hemisphere, the Cap Height (h) is equal to R, so h = 5 units.
2. Check Missing Dim (r): r = √(2Rh - h²) = √(2*5*5 - 5²) = √(50 - 25) = √25 = 5 units. (As expected, r = R for a hemisphere).
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V = (1/3) * π * (5)² * (3*5 - 5) = (1/3) * π * 25 * (15 - 5) = (1/3) * π * 25 * 10 = (250/3)π
5. Result: V ≈ 261.799 cubic units. (This is exactly half the volume of a sphere with R=5: (1/2)*(4/3)π(5³) = (2/3)π(125) ≈ 261.799).
Conclusion: The hemisphere has a volume of approx 261.8 cubic units.
Understanding Volume Measurement
Volume is the quantification of the three-dimensional space occupied by a substance or enclosed by a surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. It is measured in cubic units.
Common Volume Units Reference
Ensure your input dimensions (R, r, h) use a consistent unit (e.g., all in meters, all in inches). The resulting volume will be in the corresponding cubic units, and areas in square units.
| Linear Unit (R, r, h) | Volume Unit (V) | Area Unit (CSA, BA, TSA) | Common Equivalents |
|---|---|---|---|
| Millimeter (mm) | Cubic Millimeter (mm³) | Square Millimeter (mm²) | 1 cm³ = 1000 mm³ |
| Centimeter (cm) | Cubic Centimeter (cm³) | Square Centimeter (cm²) | 1 cm³ = 1 mL |
| Meter (m) | Cubic Meter (m³) | Square Meter (m²) | 1 m³ = 1000 Liters |
| Inch (in) | Cubic Inch (in³) | Square Inch (in²) | 1 US gallon ≈ 231 in³ |
| Foot (ft) | Cubic Foot (ft³) | Square Foot (ft²) | 1 ft³ ≈ 28.3 Liters |
| Yard (yd) | Cubic Yard (yd³) | Square Yard (yd²) | 1 yd³ = 27 ft³ |
| Kilometer (km) | Cubic Kilometer (km³) | Square Kilometer (km²) | Large scale measurements |
Frequently Asked Questions about Spherical Cap Volume
1. What is the most basic spherical cap volume formula?
The most common and fundamental formula uses the sphere radius (R) and cap height (h): V = (1/3) * π * h² * (3R - h).
2. How does this calculator find the missing dimension?
It uses the geometric relationship derived from the Pythagorean theorem: R² = r² + (R-h)². Based on which two dimensions you provide, it algebraically solves for the third one before calculating the volume and areas.
3. Can this tool calculate the volume of a hemisphere?
Yes. A hemisphere is a special case of a spherical cap where the height (h) is equal to the sphere's radius (R). Simply enter the Sphere Radius (R) and set the Cap Height (h) equal to R. The calculator will find the base radius (r), which will also equal R for a hemisphere, and calculate the volume.
4. What's the formula for the curved surface area of a spherical cap?
The area of the curved 'dome' part only is given by the formula: CSA = 2 * π * R * h. It only depends on the sphere's radius and the cap's height, not its base radius.
5. How is the total surface area calculated?
The total surface area includes the curved surface area and the flat circular base area. It's calculated as TSA = CSA + BA = 2πRh + πr². If you only need the curved area, use the CSA formula.
6. What are the valid ranges for R, r, and h?
- All dimensions must be non-negative (≥ 0).
- The cap height (h) cannot exceed the sphere's diameter (h ≤ 2R).
- The cap base radius (r) cannot exceed the sphere's radius (r ≤ R) if you are calculating the smaller cap (h ≤ R). This calculator assumes the smaller cap if you provide R and r.
- You must provide exactly two valid, non-negative numbers.
7. Why do I need to provide exactly two dimensions?
The three dimensions (R, r, h) are geometrically linked. Knowing any two allows you to uniquely determine the third, unless it's a degenerate case (like R=0). Providing only one isn't enough information, and providing three allows for inconsistencies if the numbers don't fit the geometric relationship.
8. Can this calculator handle different units?
The calculator itself works with numbers only. It's crucial that you input all known dimensions using the *same* unit (e.g., all in centimeters, or all in feet). The calculated results will then be in the corresponding cubic units (for volume) and square units (for area).
9. What is the difference between spherical cap volume and spherical segment volume?
A spherical cap is a portion of a sphere cut by a *single* plane. A spherical segment is the portion of a sphere cut by *two* parallel planes. This calculator specifically deals with the volume of a spherical cap.
10. What does it mean if the calculator gives a "Calculation Error"?
This typically means the two dimensions you entered are geometrically impossible for a real spherical cap (e.g., entering a cap base radius 'r' larger than the sphere radius 'R', or a cap height 'h' larger than the sphere diameter '2R'). Check your input values to ensure they make sense in the context of a sphere.
