Spherical Cap Volume Calculator – Finding the Volume of a Curved Dome Section
A spherical cap is the portion of a sphere that lies above (or below) a flat plane cutting through it — like the curved top of a dome, a contact lens, or the bulging end of a storage tank. Unlike a hemisphere, a spherical cap can be any height from a thin sliver to almost the entire sphere, which makes it useful for modeling a wide range of real-world curved shapes.
The Spherical Cap Volume Formula
Given the sphere's radius R and the cap's height h, the volume is:
V = (π × h² / 3) × (3R − h)
For example, with R = 10 cm and h = 4 cm: V = (π × 16 / 3) × (30 − 4) ≈ 435.74 cm³. This formula works for any cap height between 0 and 2R (the full sphere).
Using Base Radius Instead of Height
Sometimes you know the radius of the flat circular base a where the cutting plane meets the sphere, rather than the cap's height. In that case, the calculator first derives the height using:
h = R − √(R² − a²)
Once h is known, the standard volume formula applies. This dual-input approach means you never need to manually rearrange the geometry — just supply whichever measurement you have.
Curved Surface Area and Base Area
Beyond volume, the calculator also reports two useful surface measurements. The curved (lateral) surface area of the cap — the dome-shaped outer surface, excluding the flat base — is:
A_curved = 2πRh
The base circle area, the flat circular region where the cap meets the cutting plane, is:
A_base = πa²
These values are helpful when estimating material for a dome shell or the footprint of a curved cover.
Special Cases: Hemisphere and Full Sphere
The calculator automatically recognizes two special configurations. When h = R, the cap is exactly half the sphere — a hemisphere — and the volume simplifies to V = (2/3)πR³. When h = 2R, the cap spans the sphere's full diameter, meaning the "cap" is actually the entire sphere, with volume V = (4/3)πR³. Both cases are labeled clearly in the results so you always know which shape you are working with.
Unit Conversion
Inputs can be entered in mm, cm, m, in, ft, or yd. Internally, the calculator converts your radius and height to meters, computes the volume, and then presents it simultaneously in m³, cm³, mm³, liters, milliliters, in³, and ft³ — so you can instantly compare results across metric and imperial systems without doing the conversion yourself.
Step-by-Step Derivation
Enabling the step-by-step panel shows exactly how the result was reached: the formula itself, the derived height or base radius (whichever was not directly entered), the intermediate value of h² and 3R − h, and finally the substituted volume calculation. This makes the tool useful for students verifying homework or engineers double-checking a hand calculation.
Practical Applications
Spherical cap volume calculations appear in many fields:
- Tank and vessel design — estimating the liquid volume in a partially-filled spherical tank or the volume of a domed tank head
- Optics — modeling the curved surface of a lens or contact lens segment
- Architecture — computing the enclosed air volume beneath a dome roof
- Manufacturing — calculating material volume for molded dome caps, light covers, or pool covers
- Fluid dynamics — determining displaced volume when a spherical object is partially submerged
0 < h ≤ 2R, and base radius must satisfy 0 ≤ a ≤ R. Values outside these ranges are not geometrically possible and will be flagged.Accuracy
All calculations use JavaScript's native double-precision floating-point arithmetic, providing roughly 15 significant digits of accuracy. You can adjust the displayed precision from 0 to 10 decimal places to match your needs, whether that's a quick estimate or a high-precision engineering result.