Spherical Cap Calculator

Work out spherical cap instantly with clear inputs, formula shown and shareable results.

About Spherical Cap Calculator

Slicing a sphere with a plane leaves a cap of height h. Its volume is ⅓πh²(3R − h) and — remarkably — its curved area 2πRh depends only on the slice thickness, not on where the cut is made.

Formulas

Cap volume

V = ⅓πh²(3R − h)

Curved area

A = 2πRh

Base radius

a = √[h(2R − h)]

Worked example

With the default values below, Spherical Cap Calculator returns Cap volume of 435.6342. Change any input above and every figure here is recalculated in your browser.

Inputs used

Sphere radius Rminimum 0: 10
Cap height hminimum 0: 4

Results

Cap volume: 435.6342
Curved (spherical) area: 251.3274
Base circle radius a: 8.0000
Total area including the flat base: 452.3893

Step by step

1Values used
2Cap volume
3Curved area
4Base radius
5Cap volume= 435.6342
6Curved (spherical) area= 251.3274
7Base circle radius a= 8.0000
8Total area including the flat base= 452.3893

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What if h = R?

The formula gives ⅔πR³, the hemisphere, as it must.

Why is the curved area independent of position?

This is Archimedes' hat-box theorem: equal-thickness slices of a sphere have equal curved surface areas.