Spherical Cap Volume Calculator

Sphere Radius \( r \):

Cap Height \( h \):

Volume

Volume (cm³): Volume (m³): Volume (in³): Volume (ft³): Volume (liters):

Cap Base Radius \( a \)

Cap Base Radius (cm): Cap Base Radius (m): Cap Base Radius (in): Cap Base Radius (ft):

1. What is a Spherical Cap Volume Calculator?

Definition: This calculator computes the volume and cap base radius of a spherical cap (or spherical dome), which is a portion of a sphere cut off by a plane, based on the sphere's radius (\( r \)) and the cap's height (\( h \)).

Purpose: Useful in geometry, architecture (e.g., designing domes), and engineering for determining the volume and base radius of spherical cap structures.

2. How Does the Calculator Work?

The calculations are based on the following formulas:

\[ V = \frac{\pi \times h^2}{3} \times (3r - h) \] \[ a = \sqrt{r^2 - (r - h)^2} \]

Unit Conversions:

Input (Length): mm (×0.1), cm (×1), m (×100), in (×2.54), ft (×30.48) to cm
Volume (from cm³): m³ (×0.000001), in³ (×0.0610237), ft³ (×0.0000353147), liters (×0.001)
Length (from cm): m (×0.01), in (×0.393701), ft (×0.0328084)

Explanation: The inputs are converted to cm, the volume and cap base radius are calculated, and results are converted to other units.

3. Importance of Spherical Cap Volume Calculation

Details: Calculating the volume and base radius of a spherical cap is essential in architecture (e.g., designing domes), engineering (e.g., calculating the volume of tank caps), and physics (e.g., studying spherical segments).

4. Using the Calculator

Tips: Enter the sphere radius \( r \) and cap height \( h \) in mm, cm, m, in, or ft (\( r > 0 \), \( h > 0 \), and \( h \leq 2r \)). The result shows the volume in cm³, m³, in³, ft³, and liters, and the cap base radius in cm, m, in, and ft.

5. Example

Given: Sphere radius \( r = 5 \, \text{cm} \), cap height \( h = 2 \, \text{cm} \).
Volume Calculation: \( V = \frac{\pi \times 2^2}{3} \times (3 \times 5 - 2) = \frac{\pi \times 4}{3} \times 13 \approx 54.453 \, \text{cm}^3 \).
Cap Base Radius Calculation: \( a = \sqrt{5^2 - (5 - 2)^2} = \sqrt{25 - 9} = 4 \, \text{cm} \).
Conversions (Volume):
- \( \text{cm}^3 \approx 54.453 \),
- \( \text{m}^3 \approx 0.000054 \),
- \( \text{in}^3 \approx 3.323 \),
- \( \text{ft}^3 \approx 0.00192 \),
- \( \text{liters} \approx 0.054 \).
Conversions (Cap Base Radius):
- \( \text{cm} = 4 \),
- \( \text{m} = 0.04 \),
- \( \text{in} \approx 1.575 \),
- \( \text{ft} \approx 0.131 \).

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