Paraboloid Volume Calculator
Compute volume under an elliptical paraboloid from measured aperture semiaxes and height; one half of aperture area times height, not cone/cylinder volume.
Description
An elliptical paraboloid tapers with a quadratic profile.
An elliptical paraboloid tapers with a quadratic profile. Its volume is one half of aperture area times height when the aperture dimensions are semiaxes.1
Compute volume under an elliptical paraboloid from measured aperture semiaxes and height; one half of aperture area times height, not cone/cylinder volume.
When to use Paraboloid Volume Calculator
Use this tool to compute volume under an elliptical paraboloid from measured aperture semiaxes and height; one half of aperture area times height, not cone/cylinder volume.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Paraboloid Volume Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter apertureSemiAxes, height using the field table and calculation rules, then run the tool.
- Read paraboloidVolume. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| apertureSemiAxes | Aperture semi axes | list of number | Required |
| height | Height | number | Required |
Enter exactly two positive aperture semiaxes and nonnegative height in consistent length units. Multiply the elliptical aperture area by half the height; zero height gives zero volume.
volume = π × semiaxis1 × semiaxis2 × height/2 Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| paraboloidVolume | Paraboloid volume | number |
{
"input": {
"apertureSemiAxes": [
2,
3
],
"height": 4
},
"output": {
"paraboloidVolume": 37.69911184307752
}
}Limitations and assumptions
Diameters must first be halved. The formula applies to this ideal paraboloid geometry.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Ellipsoid Surface Area Approximation: Evaluate supplied power-mean ellipsoid surface-area approximation (Knud Thomsen exponent 1.6075 commonly used); exact for spheres, approximate otherwise, no implied error guarantee for arbitrary caller exponent. For a complementary task, use Bernoulli Lemniscate Area Calculator: Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.
References
-
Paraboloid Volume Calculator — tool.io
Similar or alternative tools
- Ellipsoid Surface Area Approximation
Evaluate supplied power-mean ellipsoid surface-area approximation (Knud Thomsen exponent 1.6075 commonly used); exact for spheres, approximate otherwise, no implied error guarantee for arbitrary caller exponent.
- Bernoulli Lemniscate Area Calculator
Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.