When a plane cuts a sphere, the “top piece” can be described using a height parameter \(h\).
In this calculator, \(h\) is measured from the cutting plane up to the topmost point of the sphere along the chosen reference direction.
The range is \(0 \le h \le 2r\): when \(h=r\) you get a hemisphere, and when \(h=2r\) you recover the full sphere.
Base circle radius
The cut produces a circular boundary (the base of the cap). Its radius \(a\) is:
\[
\begin{aligned}
a &= \sqrt{2rh - h^2}.
\end{aligned}
\]
This relation is purely geometric and comes from the Pythagorean theorem inside the right triangle formed by the sphere radius,
the distance from the center to the plane, and the base radius.
Spherical cap (segment) volume
The cap (also called a spherical segment) is only the sliced-off portion:
\[
\begin{aligned}
V_{\text{cap}} &= \frac{\pi h^2(3r-h)}{3}.
\end{aligned}
\]
This formula is widely used for “how much liquid is in a spherical tank above a cut” problems.
Spherical sector volume
A sector includes the cap plus the cone-like region connecting the cap boundary to the center of the sphere:
\[
\begin{aligned}
V_{\text{sector}} &= \frac{2}{3}\pi r^2 h.
\end{aligned}
\]
If you set \(h=r\), this becomes \(V_{\text{sector}}=\tfrac{2}{3}\pi r^3\), which equals the volume of a hemisphere.
Cap surface and base areas
The curved surface area of a cap (excluding the flat base) is:
\[
\begin{aligned}
A_{\text{curved}} &= 2\pi r h.
\end{aligned}
\]
The flat base area is the area of the circle of radius \(a\):
\[
\begin{aligned}
A_{\text{base}} &= \pi a^2.
\end{aligned}
\]
Visualization note: a plane can be tilted in many ways while still intersecting the sphere in a circle. The calculator’s 3D view lets you tilt the plane to build intuition,
while the formulas use the standard height-based definitions shown above.
