Spherical Cap Surface Area Calculator
From the sphere radius and the cap height, get the curved dome surface, the radius of the flat base, and the total surface area of a spherical cap.
Two inputs, three answers
Enter the sphere radius and the cap height and the calculator returns the curved surface (2πRh), the base radius (√(h(2R−h))), and the total surface at once.
Cap height cannot exceed 2R
The cap height h is measured from the flat base to the top of the dome; it can be at most the sphere's diameter (2R). The radius and height are unit-agnostic — keep one unit throughout.
What is a spherical cap surface area calculator?
Sphere radius and cap height in, full cap out
A spherical cap is the rounded piece you get when a single flat plane slices through a sphere — a dome. A spherical cap surface area calculator turns two measurements — the radius R of the original sphere and the height h of the cap — into the numbers that describe that dome: the area of its curved outer shell (curved surface), the radius of the flat circular cut at its base (base radius), and the combined area of the dome plus that base (total surface). These are the figures you need for dome roofs, contact-lens and lens-cap shapes, the wetted area of a partly filled tank, and any geometry problem where a slice of a sphere shows up.
Enter the sphere radius and the cap height in any length unit to get the curved surface, base radius, and total surface instantly.
Two short formulas, both built from the sphere radius R, the cap height h, and the constant π (about 3.14159).
curved surface = 2 × π × R × hThe curved (dome) surface is a strikingly simple 2 × π × R × h — it depends only on the sphere radius and the cap height, not on where the slice sits. The base radius a, the radius of the flat circular cut, is √(h × (2R − h)) by the chord geometry of the slice. The total surface adds the flat base circle to the dome: total = 2πRh + π × a².
Suppose you slice a sphere of radius 5 and the cap is 2 high.
Base radius
√(2 × (2 × 5 − 2)) = √(2 × 8) = √16 = 4 — the radius of the flat circular cut.
Curved surface
2 × π × 5 × 2 = 62.831853 square units — the dome's outer shell.
Total surface
62.831853 + π × 4² = 62.831853 + 50.265482 = 113.097336 square units.
The three outputs answer three different everyday questions. The curved surface (about 62.831853 square units for R = 5, h = 2) is the area of the dome's outer shell — the material to skin a dome roof, the coating on a contact lens, the wetted wall of a tank filled to depth h. The most useful insight is how simple 2πRh is: the dome area grows in direct proportion to the cap height, so a cap twice as tall has twice the curved surface, and a full hemisphere (h = R) gives 2πR², exactly half a sphere. The base radius (4 here) is the radius of the flat circular face left by the cut — the opening of the dome; note it is largest at the equator (h = R) and shrinks back to zero as the cap grows toward a full sphere. The total surface (about 113.097336) closes the dome with that flat base, the figure you want when the cap is a solid lid rather than an open shell. Use the curved surface for an open dome and the total surface for a closed one.
The formulas are exact, but a couple of practical points are worth keeping in mind.
A single flat slice and consistent units
These formulas describe a cap cut from a perfect sphere by one flat plane, so the cap height h must be between 0 and the full diameter 2R — a larger h is geometrically impossible and the calculator returns no result. A cap cut by a curved surface, an ellipsoid slice, or a frustum (a band between two parallel cuts) will differ. The radius and height are also unit-agnostic, so keep one unit throughout: a radius and height in centimetres give surfaces in square centimetres, never a mix.