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Sphere and Spherical Cap Volumes in Separate Sections

Sphere and Spherical Cap Volumes in Separate Sections. Use unit controls and documented positive geometric roots to compare base, lateral and total measures.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Radius (r) 3, Sphere radius (r) 3, Cap height (h) 1 gives Circumference 18.849556, Volume 113.09734, Cap base radius (a) 2.236068, Volume 8.3775804.

Vsphere = 4πR³/3; Vcap = πh²(3R−h)/3; acap² = h(2R−h)

Worked example

Enter these known values and leave the other values blank.

Radius (r)
3 cm
Sphere radius (r)
3 cm
Cap height (h)
1 cm
Circumference
18.85 cm
Volume
113.1 cm³
Cap base radius (a)
2.236 cm
Volume
8.378 cm³

Assumptions and limitations

  • The sphere and cap sections use independent sphere radii. A cap height lies strictly between zero and the sphere diameter. Both volumes are output-only in the captured workflow; the base-circle inverse selects the larger cap height when its radius alone does not distinguish two caps.
  • Select compatible units and enough independent dimensions. Clear unknown values before solving backward. Numeric precision and display rounding do not quantify uncertainty in measured inputs.
  • Arithmetic uses 64 significant digits. Original reference diagrams explain the dimensions and do not follow entered proportions.

Common questions

Can I solve backward by editing the volume result?

This workflow keeps both volume results output-only. Use radius, circumference and the editable cap dimensions; base-circle radius with sphere radius selects the larger cap-height branch.

References

The US fluid ounce uses one 128th of the exact US liquid gallon, replacing the rounded captured reciprocal factor. The US customary cup uses one sixteenth of the exact US liquid gallon, replacing the rounded captured cup volume. Bookify replaces the captured rounded US gallon with the exact 3.785411784-liter definition.

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