About

A bowl segment (spherical cap) is the portion of a sphere cut by a plane. Miscalculating its volume leads to material waste in tank fabrication, incorrect dosing in pharmaceutical vessel design, and structural errors in dome engineering. This calculator solves the full geometry from any two known parameters: sphere radius R, cap height h, or base radius a. It uses the exact closed-form relation a² = 2Rh − h² to derive the missing dimension, then computes volume, lateral surface area, base area, and centroid location. Results assume a geometrically perfect sphere. Real-world deviations from sphericity (common in spun or pressed metal bowls) introduce error on the order of 1−3%.

Formulas

The spherical cap is fully determined by the constraint linking sphere radius R, base radius a, and height h:

a² + (R − h)² = R²

Expanding gives the fundamental relation:

a² = 2Rh − h²

Volume is derived by integrating circular disc areas from the base plane to the apex:

V = πh²3(3R − h)

Curved (lateral) surface area equals the zone of a sphere of height h. Archimedes proved this equals the area of the corresponding cylinder wrapper:

S_lat = 2πRh

Centroid of the solid cap measured from the base plane:

ȳ = h(4R − h)4(3R − h)

Where R = sphere radius, a = base (opening) radius, h = cap height (sagitta), V = volume, S_lat = curved surface area, ȳ = centroid height from base.

Reference Data

Parameter	Formula	Unit	Notes
Sphere Radius	R = a² + h²2h	length	Derived when a and h are known
Base Radius	a = √2Rh − h²	length	Requires h ≤ 2R
Cap Height	h = R − √R² − a²	length	Requires a ≤ R (minor cap)
Volume	V = πh²3(3R − h)	length³	Exact for perfect sphere
Volume (alt)	V = πh6(3a² + h²)	length³	Useful when R unknown
Curved Surface Area	S_lat = 2πRh	length²	Lateral area only
Base Area	A_base = πa²	length²	Circular opening area
Total Surface Area	S_total = S_lat + A_base	length²	Cap + base disc
Centroid Height	ȳ = h(4R − h)4(3R − h)	length	From base plane upward
Arc Length (profile)	L = 2R arcsin(aR)	length	Cross-section arc
Hemisphere check	h = R	-	V = 23πR³
Full Sphere check	h = 2R	-	V = 43πR³
Solid Angle	Ω = 2π(1 − R − hR)	sr	Steradians subtended
Sagitta	Same as h	length	Optics terminology for cap height
Ratio h/R = 0.1	V ≈ 0.029πR³	-	Shallow cap approximation
Ratio h/R = 0.5	V ≈ 0.654R³	-	Quarter-sphere depth

Frequently Asked Questions

When h = R, the cap becomes an exact hemisphere. The base radius equals the sphere radius (a = R), volume reduces to 23πR³, and curved surface area becomes 2πR². The calculator detects this case automatically.

Yes. When h > R, you have a major spherical cap (more than a hemisphere). All formulas remain valid for h up to 2R (the full sphere). The base radius a decreases as h approaches 2R, reaching zero at the full sphere. When using the (R, a) input mode, the calculator returns the minor cap by default (h ≤ R).

Archimedes' Hat-Box Theorem proves that the area of a spherical zone depends only on the sphere radius R and the zone height h, not on where the zone sits on the sphere. Two caps of equal height on the same sphere have identical curved surface areas regardless of their base radii. The formula S_lat = 2πRh is equivalent to the lateral area of a cylinder of radius R and height h.

In pressure vessel heads (ASME code), torispherical and hemispherical caps use these formulas for internal volume and material surface calculations. For spun metal bowls, the curved surface area determines blank disc size before forming. Dome builders use the base radius and height to verify structural rise-to-span ratios. Typical tolerance in pressed steel is 1−3% deviation from perfect sphericity, so apply that margin to calculated values.

The centroid ȳ gives the center of mass location (assuming uniform density) measured from the base plane. This is critical for stability analysis of dome structures, liquid center-of-gravity in partially filled spherical tanks, and moment calculations in structural engineering. For a hemisphere, ȳ = 3R8.

The profile arc length L = 2R arcsin(a÷R) is exact for the minor cap (h ≤ R). For a major cap (h > R), the arc extends past the equator and the formula becomes L = 2R(π − arcsin(a÷R)). The calculator handles both cases.