A spherical cap is the portion of a sphere cut off by a plane.
Use the calculator below to compute its volume, lateral surface area, base radius,
and the angle α (in degrees). Select the desired input mode from the dropdown.
units: linear in "units", area in "square units", volume in "cubic units". Ratios are percentages.
Given sphere radius \( R \) and cap height \( h \) (with \( 0 < h \le 2R \)):
\[ \text{Volume}_{cap} = \frac{\pi}{3}\,h^2\,(3R - h) \quad \text{(cubic units)} \]
\[ \text{Lateral area}_{cap} = 2\pi R h \quad \text{(square units)} \]
Base radius: \( r = \sqrt{R^2 - (R-h)^2} \) (units)
Angle \( \alpha \) (from vertical axis to cap edge):
\[ \alpha = \begin{cases} \arcsin\left(\dfrac{r}{R}\right) & \text{if } h \le R \text{ (cap ≤ hemisphere)} \\ 180^\circ - \arcsin\left(\dfrac{r}{R}\right) & \text{if } h > R \text{ (cap > hemisphere)} \end{cases} \]
Ratios (percent): \( \dfrac{V_{cap}}{V_{sphere}} \times 100 \) \( \dfrac{A_{cap}}{A_{sphere}} \times 100 \)
If volume \( V \) is known, height \( h \) solves \( \frac{\pi}{3}h^2(3R-h) = V \). The engine uses a cubic solver (iterative).
* area is lateral surface (curved part).