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) - Given Sphere Radius (R) and Base Radius (r): (Assuming h ≤ R)
h = R - √(R² - r²)(Requires R ≥ r)
Spherical Cap Surface Area Formulas
- Curved Surface Area (CSA): The area of the curved 'dome' part only.
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 and base areas.
TSA = CSA + BA = 2πRh + πr²
10 Calculation Examples
Click on an example to see the step-by-step calculation.
Example 1: Dome Roof Volume
Scenario: Estimate the volume inside a dome roof.
1. Inputs: Sphere Radius (R) = 20 m, Cap Height (h) = 5 m.
2. Calculated Dim (r): r = √(2Rh - h²) = √(2*20*5 - 5²) ≈ 13.23 m.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (5)² * (3*20 - 5) = (1/3) * π * 25 * 55
5. Result: V ≈ 1,439.9 m³.
Example 2: Contact Lens Volume
Scenario: Model a contact lens as a thin spherical cap.
1. Inputs: Base Radius (r) = 6 mm, Height (h) = 1 mm.
2. Calculated Dim (R): R = (h² + r²) / (2h) = (1² + 6²) / (2*1) = 18.5 mm.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (1)² * (3*18.5 - 1) = (1/3) * π * 54.5
5. Result: V ≈ 57.1 mm³.
Example 3: Liquid in a Spherical Bowl
Scenario: A hemispherical bowl is filled with water.
1. Inputs: Sphere Radius (R) = 10 cm, Water Depth/Height (h) = 4 cm.
2. Calculated Dim (r): r = √(2Rh - h²) = √(2*10*4 - 4²) = 8 cm.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V = (1/3) * π * (4)² * (3*10 - 4) = (1/3) * π * 16 * 26
5. Result: V ≈ 435.6 cm³ (or 435.6 mL).
Example 4: Planetary Ice Cap Volume
Scenario: Estimate a planet's polar ice cap volume.
1. Inputs: Planet/Sphere Radius (R) = 3000 km, Ice Cap Base Radius (r) = 1000 km.
2. Calculated Dim (h): h = R - √(R² - r²) = 3000 - √(3000² - 1000²) ≈ 171.6 km.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (171.6)² * (3*3000 - 171.6)
5. Result: V ≈ 2.72 × 10⁸ km³.
Example 5: Cut Section of a Ball
Scenario: A solid ball is cut, removing a cap.
1. Inputs: Sphere Radius (R) = 5 in, Height of cut (h) = 2 in.
2. Calculated Dim (r): r = √(2Rh - h²) = √(2*5*2 - 2²) = 4 inches.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V = (1/3) * π * (2)² * (3*5 - 2) = (1/3) * π * 4 * 13
5. Result: V ≈ 54.45 in³.
Example 6: Liquid in Spherical Tank
Scenario: A spherical tank is partially filled with liquid.
1. Inputs: Tank/Sphere Radius (R) = 2 m, Liquid Depth (h) = 0.5 m.
2. Calculated Dim (r): r = √(2Rh - h²) = √(2*2*0.5 - 0.5²) ≈ 1.323 m.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V = (1/3) * π * (0.5)² * (3*2 - 0.5) = (1/3) * π * 0.25 * 5.5
5. Result: V ≈ 1.44 m³.
Example 7: Architectural Dome Feature
Scenario: A small decorative dome is part of a larger design.
1. Inputs: Base Radius (r) = 1 m, Height (h) = 0.3 m.
2. Calculated Dim (R): R = (h² + r²) / (2h) = (0.3² + 1²) / (2*0.3) ≈ 1.817 m.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (0.3)² * (3*1.817 - 0.3)
5. Result: V ≈ 0.485 m³.
Example 8: Optical Lens Surface
Scenario: Model the curved part of a simple optical lens.
1. Inputs: Sphere Radius (R) = 50 mm, Lens Base Radius (r) = 10 mm.
2. Calculated Dim (h): h = R - √(R² - r²) = 50 - √(50² - 10²) ≈ 1.01 mm.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (1.01)² * (3*50 - 1.01)
5. Result: V ≈ 159.7 mm³.
Example 9: Machined Spherical Indent
Scenario: Material removed when creating a spherical cap indentation.
1. Inputs: Sphere Radius (R) = 3 cm, Indent Base Radius (r) = 2 cm.
2. Calculated Dim (h): h = R - √(R² - r²) = 3 - √(3² - 2²) ≈ 0.764 cm.
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V ≈ (1/3) * π * (0.764)² * (3*3 - 0.764)
5. Result: V ≈ 5.03 cm³.
Example 10: Hemisphere Volume (Special Case)
Scenario: Calculate the volume of a hemisphere.
1. Inputs: Sphere Radius (R) = 5 units, Cap Height (h) = 5 units.
2. Calculated Dim (r): r = √(2Rh - h²) = √(2*5*5 - 5²) = 5 units. (As expected, r=R).
3. Formula (Volume): V = (1/3)πh²(3R - h)
4. Calculation: V = (1/3) * π * (5)² * (3*5 - 5) = (250/3)π
5. Result: V ≈ 261.8 cubic units.
Frequently Asked Questions
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 with only two dimensions?
It uses the geometric relationship between R, r, and h (derived from the Pythagorean theorem) to calculate the missing third dimension first. Then it uses the primary volume formula to find the cap's properties.
3. What's the difference between a spherical cap and a spherical segment?
A spherical cap is cut from a sphere 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. For a hemisphere, the cap height (h) is equal to the sphere radius (R).
5. How is the Curved Surface Area (CSA) calculated?
The area of the curved part of the cap is calculated with the formula CSA = 2 * π * R * h. It depends on the sphere radius and cap height, not the base radius.
6. What are the limitations on the input values?
- All values (R, r, h) must be positive or zero.
- The cap height (h) cannot be greater than the sphere diameter (2R).
- The cap base radius (r) cannot be greater than the sphere radius (R).
- The calculator requires exactly two valid dimensions to perform a calculation.
7. What units should I use for the calculation?
You can use any unit (e.g., cm, meters, inches, feet), but you must be consistent for all inputs. The output volume will be in cubic units (e.g., cm³), and the areas will be in square units (e.g., cm²).
8. Can I use this to find the volume of a dome?
Yes. Many architectural domes are shaped like spherical caps. You can model a dome by entering its height and base radius (or overall sphere radius) to find its approximate volume and surface area.
9. How do I calculate the volume of the remaining part of the sphere?
First, calculate the volume of the full sphere using the formula Vsphere = (4/3) * π * R³. Then, calculate the volume of the cap (Vcap) with this tool. The remaining volume is Vremaining = Vsphere - Vcap.
10. If I only know the sphere radius (R) and base radius (r), how is the height (h) found?
The calculator finds the height of the smaller possible cap using the formula h = R - √(R² - r²). This is derived from the Pythagorean theorem.
