Bowl Segment Calculator

A bowl segment is modeled here as a spherical cap: the portion of a sphere cut by a plane. This shape appears in bowls, domes, dished vessel heads, landscape basins, architectural shells, and turned wood projects. Two dimensions define the geometry: the radius of the complete parent sphere and the vertical height of the segment measured from the cutting plane to the cap’s highest or lowest point. The cap volume equals pi multiplied by the height squared and by three times the sphere radius minus the height, all divided by three. The curved surface area equals two times pi times the sphere radius times the height. The opening or base radius follows from the right-triangle relationship and equals the square root of two times the sphere radius times the height minus the height squared. These formulas describe an ideal sphere and do not include wall thickness. A segment height equal to the sphere radius creates a hemisphere. A smaller height creates a shallow cap, while a height greater than the radius describes more than half of the sphere. The largest valid height is twice the sphere radius, which represents the complete sphere and reduces the opening radius to zero. Values beyond that limit cannot form a real spherical segment and are rejected. All input and output dimensions use the same unit system. Enter centimeters to receive cubic centimeters for volume, square centimeters for curved area, and centimeters for base radius. The same pattern works for inches, millimeters, or meters. The reported curved area excludes the flat circular opening. If a closed vessel or solid cap needs a base, add pi times the base radius squared to obtain total exposed area. Real fabricated bowls may depart from a perfect spherical profile because of seams, lips, flanges, flat bottoms, forming tolerances, or varying thickness. Material estimates should include those details plus cutting waste and joining allowance. For structural shells and pressure vessels, geometry is only one part of design; loads, material properties, supports, buckling, weld details, and governing codes also matter. Use this calculator for geometry and preliminary quantity planning, then confirm production dimensions from the final drawing or model.