Spherical Cap Volume CalculatorFree online math calculator — instant results, no sign-up
Calculate a spherical cap's volume from sphere radius and cap height.
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About the Spherical Cap Volume Calculator
The Spherical Cap Volume Calculator finds the volume of the portion of a sphere cut off by a flat plane using the sphere radius and cap height. Enter these two lengths in the same unit to obtain the cap volume in cubic units. With the default radius of 5 and cap height of 2, the result is 54.4542726622 cubic units.
- Sphere radius
- The distance from the sphere’s center to its surface, entered in a length unit such as centimeters or meters.
- Cap height
- The perpendicular distance from the cap’s flat cutting plane to its farthest point on the spherical surface, entered in the same length unit as the sphere radius.
Formula and method
V = (πh²/3)(3R − h); with R = 5 and h = 2, V = (π × 2²/3)(3 × 5 − 2) = 52π/3 ≈ 54.4542726622.
- VSpherical cap volume, in cubic units.
- RRadius of the original sphere, in length units.
- hCap height measured perpendicular to the cutting plane, in the same length units as R.
- πThe mathematical constant pi, approximately 3.141592653589793.
Square the cap height, multiply by three times the sphere radius minus the cap height, and multiply by π/3. For the default inputs, h² = 4 and 3R − h = 13, giving V = 52π/3 ≈ 54.4542726622 cubic units.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Sphere radius5
- Cap height2
Results
- Result54.4542726622
How to use the Spherical Cap Volume Calculator
- 1Enter the sphere radius in “Sphere radius”; the default value is 5.
- 2Enter the cap’s perpendicular height in “Cap height”; the default value is 2.
- 3Check that both lengths use the same unit, the radius is positive, and the height lies between zero and twice the radius.
- 4Run the calculation to apply V = (πh²/3)(3R − h).
- 5Read the Result row as a volume in cubic units; the default inputs produce 54.4542726622.
Reading your result
The Result row gives the volume enclosed by the curved spherical surface and the flat circular base. For radius 5 and cap height 2, Result = 54.4542726622 cubic units. If the inputs are in centimeters, this means cubic centimeters; if they are in meters, it means cubic meters.
Limitations
The formula assumes a perfect sphere cut by a flat plane and calculates enclosed volume, not surface area or the material volume of a hollow shell. Valid geometric inputs require R > 0 and 0 ≤ h ≤ 2R; values outside this range do not describe a spherical cap, even if the expression returns a number. Both inputs must use the same length unit. The displayed decimal is an approximation, and its precision does not imply equally precise measurements.
Common questions
Is “Sphere radius” the radius of the cap’s circular base?+
No. It is the radius of the original sphere. The circular base radius is a different measurement, related by a² = 2Rh − h², where a is the base radius.
How should I measure the cap height?+
Measure straight from the cutting plane to the cap’s farthest point on the sphere, perpendicular to the plane. Do not measure along the curved surface.
Can the cap height be greater than the sphere radius?+
Yes. A height greater than R and no greater than 2R describes a cap containing more than half the sphere. At h = R the cap is a hemisphere, and at h = 2R it is the entire sphere.
Why do the default inputs produce 54.4542726622?+
Substituting R = 5 and h = 2 gives V = (π × 4/3) × 13 = 52π/3. Its decimal value, rounded to ten decimal places, is 54.4542726622 cubic units.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.