What Is the Spherical Cap Volume and Area Calculator?
A spherical cap (or spherical dome) is the portion of a sphere cut off by a plane.
When the cutting plane passes through the exact center of the sphere (h = R), the spherical cap forms a hemisphere.
How Does the Spherical Cap Volume and Area Calculator Work?
Base radius a is determined from Pythagorean relationship: a = √(h·(2R - h)).
Cap volume formula: V = (1/3)·π·h²·(3R - h).
Curved dome area formula: A_curved = 2·π·R·h.
Spherical Cap Volume and Area Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
Volume is computed by integrating spherical cross-sections, and curved dome surface area is exactly 2πRh.
How to Use the Spherical Cap Volume and Area Calculator
- Enter the radius R of the underlying sphere.
- Enter the height h of the cap from base to apex.
- View volume, curved surface area, circular base radius, and total enclosed area.
Step-by-Step Example Calculation
Architectural Dome (R = 10, h = 4)
Input Values:
Understanding Your Result
At h = R, the volume equals ⅔πR³ (exactly half the volume of a sphere).
The curved dome surface area depends strictly on sphere radius R and cap height h, independent of base circle radius.
Factors That Affect the Result
- Cap Height: As height approaches zero, volume scales quadratically with h.
When Should You Use This Calculator?
- Sizing geodesic architectural domes, stadium roofs, and observatory cups.
- Calculating liquid volume inside dished boiler tank heads and brewing vessels.
Assumptions & Limitations
- Cap height cannot exceed the sphere diameter (h ≤ 2R).
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard analytical calculus integration of a sphere segment.