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
10 Real-Life Spherical Cap 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, 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 ≈ 13.23 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 ≈ 1440 cubic meters.
Example 2: Contact Lens Volume
Scenario: Model a contact lens as a thin spherical cap to estimate its material volume.
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 ≈ 57.1 cubic mm.
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 ≈ 435.6 cubic cm (or 435.6 mL).
Example 4: Planetary Ice Cap Volume
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 - 2828.4 = 171.6 km.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (171.6)² * (3*3000 - 171.6) ≈ (1/3) * π * 29446 * 8828.4
5. Result: V ≈ 2.72 × 10⁸ 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 ≈ 54.45 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 ≈ 1.44 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 ≈ 1.817 meters.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (0.3)² * (3*1.817 - 0.3) = (1/3) * π * 0.09 * 5.151
5. Result: V ≈ 0.485 cubic meters.
Example 8: Optical Lens Surface
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 - √2400 ≈ 50 - 48.99 = 1.01 mm.
3. Formula (Volume): V = (1/3) * π * h² * (3R - h)
4. Calculation: V ≈ (1/3) * π * (1.01)² * (3*50 - 1.01) = (1/3) * π * 1.02 * 148.99
5. Result: V ≈ 159.7 cubic mm.
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 ≈ 1.323 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 ≈ 1.44 cubic inches.
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) = √25 = 5 units. (As expected, r = R).
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 * 10 = (250/3)π
5. Result: V ≈ 261.8 cubic units.
Frequently Asked Questions (FAQ)
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).
3. What's the difference between a spherical cap and a sphere segment?
A spherical cap is a portion of a sphere cut by a single 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 non-negative.
- 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).
- You must provide exactly two valid dimensions for the calculation to work.
7. What units should I use for the dimensions?
You can use any consistent linear units (like cm, meters, inches, feet) for all inputs. The resulting volume will be in the corresponding cubic units (cm³, m³, in³, ft³), and the areas will be in square units.
8. Is the volume of a dome the same as a spherical cap?
Yes, many architectural domes are shaped like spherical caps, so this formula is frequently used to accurately approximate their volume and surface area for construction and engineering purposes.
9. How can 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) using this tool. The remaining volume is simply 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 relationship derived from the Pythagorean theorem: h = R - √(R² - r²). This calculation assumes you want the smaller of the two possible caps (where h ≤ R).
