Spherical Cap Volume Calculator

What is a Spherical Cap?

A spherical cap (or spherical segment of one base) is a portion of a sphere cut off by a plane. It consists of a flat circular base and a curved spherical dome surface. If the cutting plane passes through the center of the sphere, the spherical cap becomes a hemisphere.

Formulas for Spherical Cap Volume

The calculator supports three mathematical methods depending on the parameters you know:

1. Given sphere radius R and cap height h:

$$V=\frac{1}{3}\pi h^2(3R-h)$$

2. Given base radius r and cap height h:

$$V=\frac{1}{6}\pi h(3r^2+h^2)$$

3. Given sphere radius R and base radius r:
The height $h$ is calculated automatically as: $$h=R-\sqrt{R^2-r^2}$$ and then the first formula is applied to find the volume.

Where:

• R — sphere radius;
• r — base radius of the cap;
• h — cap height (dome height);
• π ≈ 3.14159265;

Practical Applications

Calculating the volume of a spherical cap is critical in various industrial and scientific fields:

• Architecture & Construction: Designing domes for buildings, spherical roofs, and exhibition halls. Estimating volume helps compute material costs (like concrete), structural load weight, and the inner volume of air under the dome for HVAC systems.
• Tanks & Chemical Engineering: Most fuel containers, boilers, silo ends, and gas tanks have dome-shaped spherical heads. Cap volume allows engineers to compile accurate tank calibration tables and calculate exact fuel levels.
• Optics & Glassware: Manufacturing spherical lenses for glasses, microscopes, binoculars, and cameras. Calculating glass cap volume determines the precise weight of the optical blank before fine grinding.
• Medicine: Modeling contact lenses in ophthalmology, designing bone implants and joint cups in orthopedics, and measuring tumor volumes from imaging scans in oncology.
• Food Industry: Designing baking molds, dome-shaped lids for food containers, and mixing bowls of spherical geometry.
• Astronomy & Geodesy: Calculating planetary segment volumes when studying planetary caps, craters, or analyzing surface regions of planets and moons.

How to use the calculator?

1. Select your calculation method based on your known parameters (e.g., via R and h).
2. Enter the values in the fields.
3. Set the rounding precision.
4. Click the "Calculate" button — the result will appear instantly.

Frequently Asked Questions (FAQ)

How does a spherical cap differ from a spherical sector?
A spherical cap is just the "cut off dome" of a sphere. A spherical sector is like an ice cream cone with a rounded top: it contains the dome (cap) and a cone pointing down to the center of the sphere.

What is a full sphere as a special case of a cap?
If the height of the cap h equals the sphere diameter 2R, the cap becomes the entire sphere. In this case, the cap volume formula simplifies to the standard sphere volume formula: V = 4/3 π R³.

What units of measurement does this calculator use?
This calculator works with any consistent unit of measurement (meters, inches, millimeters, etc.). Make sure all inputs use the same unit. If you input radii in meters, the volume will be calculated in cubic meters.