Spherical Cap Calculator
Volume and curved area of a spherical cap.
Cap volume
254.469
height 3 of radius 10
Curved surface
188.4956
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Spherical Cap Calculator works
The Spherical Cap Calculator computes the volume and curved surface area of a spherical cap — the portion of a sphere sliced off by a flat plane — using just two input measurements. It's ideal for engineers, architects, students, and hobbyists working with dome structures, tank capacities, lens geometry, or any 3D application involving partial spheres.
A spherical cap is defined by two key dimensions: the radius of the full sphere (R) from which the cap is cut, and the height (h) of the cap measured perpendicularly from the flat circular base to the topmost point of the dome. Some formulations also use the base radius (a) of the circular cross-section instead of R, and the calculator can typically accept either pair of known values. Understanding which measurements you actually have in hand — sphere radius vs. base radius — is the first step to getting an accurate result.
The volume formula for a spherical cap is V = (π h² / 3)(3R − h), where R is the sphere radius and h is the cap height. This elegant expression shows that volume scales with the square of height and linearly with sphere radius, meaning even modest increases in cap height produce disproportionately large volume gains. The lateral (curved) surface area is given by A = 2πRh, which is pleasingly simple — it depends only on the sphere radius and cap height, not on the base radius directly. These formulas are derived from integration in calculus and are exact for a perfect sphere.
A common source of error is confusing the sphere radius R with the base radius a of the cap's circular footprint. These are related by the Pythagorean identity a² = R² − (R − h)² = h(2R − h), so if you know a and h but not R, you can solve for R = (a² + h²) / (2h). Another frequent mistake is measuring height incorrectly — cap height must be the perpendicular distance from the flat base to the apex, not the slant height along the curved surface. Mixing these up leads to significant overestimates of both volume and area.
Practical applications of spherical cap geometry are widespread. Architectural domes, pressure vessel end-caps, contact lenses, radio telescope dishes, and the liquid volume in a partially filled spherical tank all rely on these formulas. For tank-filling problems in particular, engineers use the cap volume formula iteratively at different fill heights h to build a volume-vs-depth table. Keep in mind that the formulas assume a geometrically perfect sphere; real-world manufacturing tolerances or ellipsoidal departures will introduce small discrepancies that may need correction for high-precision engineering work.
Formula
V = πh²(3r − h)/3
Pro tips
- If you only have the base diameter and cap height from a physical measurement, compute R first using R = (a² + h²) / (2h) before plugging into the volume or area formulas.
- To verify your inputs are geometrically consistent, check that h ≤ 2R — a cap height greater than the sphere's diameter is physically impossible.
- For dome construction or material estimation, remember the lateral surface area formula gives only the curved exterior; add the area of the circular base (πa²) if you need to cover or seal the flat face as well.
- When calculating liquid volume in a spherical tank at a given fill depth, use h as the fill height and R as the tank's sphere radius — the result is the exact volume of liquid, assuming the tank is a perfect sphere.
- Double-check your unit consistency: mixing centimeters for height with meters for radius is one of the most common errors and will produce volume results that are off by factors of 1,000 or more.
Key terms
- Spherical Cap
- — The region of a sphere that lies on one side of a cutting plane, resembling a dome shape.
- Cap Height (h)
- — The perpendicular distance from the flat circular base of the cap to its highest point (the apex).
- Sphere Radius (R)
- — The radius of the complete sphere from which the cap is geometrically derived.
- Base Radius (a)
- — The radius of the flat circular cross-section that forms the bottom edge of the spherical cap.
- Lateral Surface Area
- — The curved outer surface area of the cap, excluding the flat circular base, calculated as 2πRh.
- Hemisphere
- — A special case of a spherical cap where the cutting plane passes through the center, making h equal to R.
