Paraboloid of Revolution Volume

a dished or parabolic vessel bottom, or the shape the free surface of a spinning liquid takes. Full V = (1/2) pi R^2 H, EXACTLY half the cylinder of the same base and height (halfway between the cone's 1/3 and the cylinder's 1). A parabola runs z proportional to r^2, so filling from the apex the wetted radius grows as r(y) = R sqrt(y/H) and the liquid as V(y) = pi R^2 y^2/(2H) - it fills as the SQUARE of the depth, so a stick reading half the height holds only a quarter of the volume (percent full = (y/H)^2). A D = 4 ft, H = 3 ft dish holds 18.85 ft^3 (141.0 gal) full and 4.71 ft^3 (35.3 gal) at a 1.5 ft apex depth (25%). A circular dish is a spherical cap, a cone is a frustum with a zero top, and an off-axis or tilted paraboloid is separate. A shop and takeoff aid; verify critical dimensions on the work.

Full V = (1/2) pi R^2 H (half the enclosing cylinder); apex-up partial fill V(y) = pi R^2 y^2/(2H), with the wetted radius r(y) = R sqrt(y/H); percent full = (y/H)^2.

The paraboloid of revolution volume - standard solid geometry (Pappus's theorem / integration) as in Machinery's Handbook (Industrial Press), by name; public domain.

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

Estimate. AHJ and licensed professional govern.

Field names used by the API: base_diameter_ft, height_ft, fill_depth_ft, full_ft3, fill_ft3, percent_full

• Full volume V = (1/2) pi R^2 H, half the enclosing cylindersolid geometry (Pappus)
• Partial fill V(y) = pi R^2 y^2/(2H) from the apex, r(y) = R sqrt(y/H); fills as the square of depthintegration of the parabola z ~ r^2
• Scope solid of revolution about the axis; an off-axis or tilted paraboloid is separatescope of this tile