Spherical Cap / Dome / Partial-Fill Volume

The volume of a spherical cap - a spherical tank's partial fill, a dome, a dished tank bottom, or a hemispherical vessel. V_cap = (pi h^2/3)(3R - h) for a fill depth h from the bottom; the full sphere is (4/3) pi R^3. A 10 ft sphere filled 3 ft holds 113 ft^3 = 846 gal, only 21.6% full (not 30%, because the sphere is narrow near the bottom); fill to the 5 ft centerline for exactly half (1,958 gal), to the top for 3,917 gal. That curvature is why a spherical or dished-bottom tank needs this formula, not a straight-sided estimate. A spherical zone between two levels and an ellipsoidal head are separate. A takeoff / dipstick aid; verify against the tank chart or drawing.

R = D/2; V_cap = (pi h^2/3)(3R - h); V_full = (4/3) pi R^3; percent full = V_cap/V_full. h=R hemisphere, h=D full sphere.

The spherical-cap volume V = (pi h^2/3)(3R - h) and the full-sphere volume (4/3) pi R^3 - standard solid geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure solid geometry, public; the sphere diameter and fill depth are user-supplied measurements.

Estimate. AHJ and licensed professional govern.

Field names used by the API: sphere_diameter_ft, fill_depth_ft, cap_volume_ft3, cap_volume_gal, percent_full

• Cap volume V = (pi h^2/3)(3R - h); h from the bottom of the spheresolid geometry
• Limits h = R is a hemisphere (half), h = D the full sphere (4/3 pi R^3)solid geometry
• Scope cap only; spherical zone, ellipsoidal head, and surface area are separatescope of this tile