Spherical cap calculator
This spherical cap calculator works from the sphere radius and cap height, or from the base radius and cap height. With R = 5 and h = 2 the volume is 54.45 and the total surface area 113.10, with the calculation steps.
Example: Known values: Sphere radius R and height h · Sphere radius R 5 · Cap height h 2 → Result: Volume V = 54.45427 · Surface area S = 113.0973. Source: Wikipedia – Kugelkappe. Updated: .
How it is calculated
A spherical cap is the piece you get when a plane cuts a slice off a sphere. It has a flat circular base of radius a and a curved top of height h. Domes, watch glasses and lenses are typical examples.
The formulas
- Base radius: a = √(h · (2R − h))
- Volume: V = π · h² · (3R − h) ÷ 3
- Curved surface area: K = 2 · π · R · h
- Total surface area: S = K + π · a²
- Sphere radius from a and h: R = (a² + h²) ÷ (2h)
How to use it
Choose the known measures and enter the values. Example: R = 5 and h = 2 give a = 4, V = π · 4 · 13 ÷ 3 = 54.45, K = 62.83 and S = 62.83 + 50.27 = 113.10. With h = 2R you get the whole sphere, with h = R a hemisphere. The unit is up to you; areas are in its square, the volume in its cube.
Frequently asked questions
How do you calculate the volume of a spherical cap?
V = π · h² · (3R − h) ÷ 3. For R = 5 and h = 2 that is π · 4 · 13 ÷ 3 = 54.45.
How do you calculate the curved surface area?
K = 2 · π · R · h, the curved part only. For R = 5 and h = 2 that is 62.83. The flat base adds to the total surface area.
How do you find the sphere radius from the base and height?
R = (a² + h²) ÷ (2h). Choose “Base radius a and height h”; for a = 4 and h = 2 you get R = 5.
What is the difference between a cap and a segment?
Both describe the same piece; “cap” often means just the curved surface. Two parallel cuts produce a spherical zone instead.
What happens at h = R?
The cap is a hemisphere: V = 2/3 · π · R³ and a = R.
Sources and legal basis
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