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Spherical cap: volume and surface area calculator

Sizes

    How to calculate the volume and surface area of a spherical cap

    The part of a sphere of radius r cut off by a plane, h tall: a dome, a contact lens, the top of an orange. Archimedes again: its curved surface is 2πrh, as if you unrolled a cylinder band of the same height. At h = r it is a hemisphere, at h = 2r the whole sphere.

    Formulas

    Base radius
    ρ = √(h(2r - h))
    Volume
    V = (π h² (3r - h))/3
    Curved surface
    S_c = 2π r h
    Base disc
    S_b = π ρ²
    Surface area
    S = S_c + S_b

    Worked example

    Given: sphere radius r = 4 cm, cap height h = 2 cm.

    1. Base radiusρ = √(2 · (2 · 4 - 2)) = 3.4641 cm
    2. VolumeV = (π · 2² · (3 · 4 - 2))/3 = 41.8879 cm³= 0.0419 l
    3. Curved surfaceS_c = 2 · π · 4 · 2 = 50.2655 cm²
    4. Base discS_b = π · 3.4641² = 37.6991 cm²
    5. Surface areaS = 50.2655 + 37.6991 = 87.9646 cm²

    Common questions about the spherical cap

    How do you calculate the volume of a spherical cap?

    Use V = (π h² (3r - h))/3. For example, with r = 4 cm and h = 2 cm, V = 41.8879 cm³.

    How do you calculate the surface area of a spherical cap?

    Use S = S_c + S_b. For example, with r = 4 cm and h = 2 cm, S = 87.9646 cm².

    Can I find the sphere radius from the volume?

    Yes. In Solve, choose the volume under “I know”, type its value and choose the sphere radius under “Find”. The other sizes stay as they are. For example, with h = 2 cm and V = 100 cm³, r = 8.6244 cm.

    Related shapes

    All 37 shapes and their main formulas

    Each one has its own model, sizes you can change and the formula behind every number. Pick one to open it in the lab.

    How the lab works

    Change a size, watch everything follow

    Drag a slider or type an exact value. The model, every result and every worked formula update at once, and the grid under the model counts one unit per square.

    See what a number means

    Tap a result and the model shows it: water fills a volume, paint sweeps a surface or an area, a dot walks the perimeter, vertices and edges are counted one by one.

    Work backwards

    Know the area and need the radius? Solve finds the size that gives the value you typed, then shows every other measurement for it.

    One set of formulas everywhere

    Every formula is written once and evaluated by the same engine in this page and on the API, with a test that checks both give the same numbers.

    Questions

    Can I work backwards from an area or a volume?

    Yes. Open Solve, choose the measurement you know, type its value and pick the size to find. The lab searches for the size that gives exactly that value and shows every other measurement for it, for example a circle with an area of 200 cm² has a radius of 7.9788 cm and a circumference of 50.1326 cm.

    How accurate are the results?

    Every formula is exact and computed in double precision, then rounded to the number of decimals you choose. Two values have no exact elementary formula and are labelled as approximations: the perimeter of an ellipse (Ramanujan) and the surface of an ellipsoid (Knud Thomsen, within 1.07 %).

    Which units can I use?

    Millimetres, centimetres, decimetres, metres, kilometres, inches, feet and yards. Areas and volumes follow the length unit (cm², cm³), and every volume is also given in litres.

    Is there an API?

    Yes. The same engine answers at https://techtools.cz/geometry-lab-api/ without a key: list the shapes, compute any of them, solve backwards, or evaluate an expression. The documentation is at https://techtools.cz/geometry-lab-api/docs.

    For developers and AI agents

    Public JSON, no key, no sign-in. Every response carries the formula, the worked step and the value.

    curl 'https://techtools.cz/geometry-lab-api/compute?shape=cone&r=3&h=4&unit=cm'
curl 'https://techtools.cz/geometry-lab-api/solve?shape=circle&given=A&value=200&unknown=r'
curl 'https://techtools.cz/geometry-lab-api/calc?expr=2pi*r^2&vars%5Br%5D=3'
curl 'https://techtools.cz/geometry-lab-api/shapes'