Spherical Cap Volume Calculator

Geometry behind spherical caps

A spherical cap appears whenever a plane slices through a sphere, leaving a domed section of height atop a sphere of radius . Architects meet the shape in domed roofs, tank designers evaluate it when sizing storage vessels, and scientists apply the same math to ice sheets or droplets. Once you know and , the remaining dimensions follow from classic integral geometry.

The circular footprint formed by the slicing plane has radius . Integrating the areas of infinitesimally thin disks stacked through the cap yields the familiar volume relation:

The curved surface area follows a linear relationship with height: . Because the same unit carries through every result, you can mix and match meters, inches, or centimeters as long as the radius and height share the system.

Scenario comparisons

Use the table to benchmark the scale of typical caps, from architectural domes to satellite ice sheets.

Notice how the volume grows rapidly with height; a modest increase in can double storage needs. The area term expands linearly, which helps painters or materials engineers estimate surface finishing requirements.

Design tips and next steps

Real-world caps seldom match theoretical perfection. Construction tolerances, sagging materials, or uneven supports can shift measurements, so verify both radius and height in the field. When heights approach the diameter of the sphere, sensitivity to measurement error increases dramatically. In those cases, sample multiple points and average them before computing a final design volume.

Continue planning curved structures with the Sphere Volume and Surface Area Calculator, verify material usage with the Geodesic Dome Strut Length Calculator, and estimate related fluid behavior with the Torricelli Tank Draining Time Calculator.