3D Spherical Segment Calculator

This online tool computes geometric properties of a spherical segment — either a spherical cap cut by one plane or a truncated segment between two parallel planes. Enter the sphere radius and either heights or an angle, choose units, and the calculator returns volume, surface areas, and mass for a solid or hollow shell given a material density and wall thickness. Results are presented numerically and visualized in 3D for quick verification. The tool runs entirely in the browser on phones, tablets and desktops.

What you can compute

• All main parameters of a spherical cap: cap height, base radius, cut angle, base area, curved area and total surface area.
• Geometry of a truncated spherical segment between two parallel cuts: segment height, radii of the two circular faces, areas and volume.
• Mass of a solid body from a given density in kg/m³.
• Mass and material volume of a hollow shell with specified wall thickness.
• Automatic unit handling for millimetres, centimetres and metres.
• Automatic conversion between cap height and cut angle when an angle is entered.
• Formatted numeric output with user-selected rounding and an interactive 3D preview for visual inspection.

Notation and core formulas

Spherical cap, single plane cut

Symbols:

• \(R\) — sphere radius
• \(h\) — cap height measured from the cutting plane to the sphere apex
• \(\theta\) — cut angle in radians or degrees in the UI, measured from the sphere axis to the plane
• \(a\) — radius of the circular base (cut circle)

Height from angle:

$$h = R(1 – \cos\theta)$$

Base radius:

$$a = \sqrt{2Rh – h^{2}}$$

Areas:

Base area: \(\;A_{\text{base}} = \pi a^{2}\)

Curved (lateral) area: \(\;A_{\text{curve}} = 2\pi R h\)

Total exterior surface: \(\;A_{\text{total}} = A_{\text{base}} + A_{\text{curve}}\)

Volume (cap):

$$V = \frac{\pi h^{2}}{3}\,(3R – h)$$

Truncated segment, between two parallel planes

Symbols:

• \(h_{1}, h_{2}\) — distances from the sphere apex to the lower and upper cutting planes (with \(0 \le h_{1} < h_{2} \le 2R\))
• \(a_{1}, a_{2}\) — radii of the lower and upper circular faces at heights \(h_{1}\) and \(h_{2}\)
• \(h = h_{2} – h_{1}\) — height of the truncated segment

Face radii:

$$a_{1} = \sqrt{2Rh_{1} – h_{1}^{2}},
$$
$$
a_{2} = \sqrt{2Rh_{2} – h_{2}^{2}}$$

Areas of faces:

$$A_{1} = \pi a_{1}^{2},\qquad A_{2} = \pi a_{2}^{2}$$

Curved lateral area for the segment:

$$A_{\text{curve}} = 2\pi R h$$

Total exterior surface:

$$A_{\text{total}} = A_{1} + A_{2} + 2\pi R h$$

Volume of the truncated segment:

$$V = \frac{\pi h}{6}\,\bigl(3a_{1}^{2} + 3a_{2}^{2} + h^{2}\bigr)$$

Mass and hollow shells

If density \(\rho\) (kg/m³) is given and volume \(V\) is in m³, the solid mass is

$$m = \rho\,V$$

For a hollow shell with wall thickness \(t\), the inner radius is

$$R_{\text{inner}} = R – t$$

Compute internal heights and volumes with \(R_{\text{inner}}\) and subtract the inner volume from the outer volume:

$$m_{\text{hollow}} = \rho\,(V_{\text{outer}} – V_{\text{inner}})$$

How the tool works and input rules

• All internal calculations use SI length units, millimetres converted to metres internally, to preserve numeric stability; the interface accepts mm, cm and m and converts automatically.
• For the cap you may supply either the height \(h\) or the cut angle \(\theta\); entering one updates the other.
• For a truncated segment provide two heights \(h_{1}\) and \(h_{2}\); the calculator enforces \(0 \le h_{1} < h_{2} \le 2R\).
• When a hollow shell is requested the thickness must be positive and strictly less than the sphere radius; inner geometry is computed and subtracted.
• Results are rounded according to the user-specified number of decimal places and presented in the chosen length units.
• 3D preview reflects the selected geometry and thickness; for large models the preview uses optimized tiling to keep interactivity smooth.

Validation and practical notes

• Check units before entering values — a misplaced metre vs millimetre will produce large errors.
• If the computed inner radius becomes zero or negative the thickness value is invalid and must be reduced.
• Surface area formulas include the circular faces; if you only need curved area use the curved area expression \(2\pi R h\).
• Use density values from supplier data for accurate mass estimates; default density may be a placeholder.

Typical parameter table

R (mm)	h (mm)	a (mm)	V (cm³)	Curved area (cm²)	Base area (cm²)
80	10	39.20	33.51	50.27	4.83
120	25	72.11	221.67	188.50	16.35
150	45	123.69	887.92	424.12	48.10
200	60	171.47	2261.95	754.77	92.50
300	120	265.83	10179.56	2261.95	222.02

✍ Outputs are suitable for engineering estimates, prototyping and material planning. Numerical results are computed from closed-form formulas; for manufacturing or structural calculations validate results with CAD, finite element analysis or direct measurement as required. The hollow-shell mode approximates a uniform wall thickness; complex variable-thickness shells are outside the scope of this calculator.

This calculator is provided for convenience and planning. Use results at your own risk. If you reuse formulas or code, please credit Pro Calculators Lab and link back to the tool.

References

1. Geometry and the Imagination, David Hilbert & S. Cohn-Vossen, AMS Chelsea Publishing, 1999.
2. Solid Geometry with Applications and Problems, H.S. Hall & F.H. Stevens, Macmillan, 1912.
3. Mathematics for Engineers and Scientists, Alan Jeffrey, Chapman & Hall/CRC, 2010.
4. Geometry of Surfaces, John Stillwell, Springer, 1992.