Spherical Cap Calculator (Volume and Surface Area of a Dome)
Volume, base radius, curved and total surface area of a spherical cap — a dome, a bowl, a bubble cut by a plane — from the sphere's radius and the cap's height.
Results
The dashed line marks the value you entered.
What this tool does
A spherical cap is what is left when a plane slices off the top of a ball: a dome with a flat circular base. Its volume is π h² (3R − h) ÷ 3, and the curved area is the surprisingly simple 2π R h — exactly the same as the band of a cylinder wrapped round the sphere with the same height, a result Archimedes found. When the height equals the radius you have a hemisphere, and at twice the radius the whole sphere, which is a quick check on any answer. The height is measured from the flat face to the top of the dome, not along the surface.
Formula
V = π h² (3R − h) ÷ 3 · a = √(h(2R − h)) · curved area = 2π R h · total = 2π R h + π a²
Variables
| Symbol | Meaning | Unit |
|---|---|---|
rs | Radius of the sphere | — |
h | Height of the cap | — |
V | Volume of the cap | — |
AB | Radius of the flat base | — |
AC | Curved surface area | — |
AT | Curved area plus flat base | — |
TH | Half-angle seen from the centre | ° |
SH | Share of the whole sphere's volume | % |
Worked example
- Radius of the sphere10
- Height of the cap4
- Volume of the cap435.634
- Radius of the flat base8.000
- Curved surface area251.327
- Curved area plus flat base452.389
- Half-angle seen from the centre53.13 °
- Share of the whole sphere's volume10.40 %
Limitations
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.