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Circular Frustum Volume with Perpendicular and Slant Height

Circular Frustum Volume with Perpendicular and Slant Height. Convert compatible units, recover unknown dimensions and distinguish curved surfaces, bases 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: Base radius (R) 3, Top radius (r) 1, Height (h) 4 gives Slant height (s) 4.472136, Volume 54.454273.

V = πh(R²+Rr+r²)/3; s² = h²+(R−r)²

Worked example

Enter these known values and leave the other values blank.

Base radius (R)
3 cm
Top radius (r)
1 cm
Height (h)
4 cm
Slant height (s)
4.472 cm
Volume
54.45 cm³

Assumptions and limitations

  • A right circular frustum has two parallel circular ends. Height is perpendicular to the ends. Equal radii produce a cylinder. Both radii must be positive in this workflow. A radius inferred from slant and height selects the larger-radius branch; volume and a known radius recover the positive volume-based root.
  • Select units before entering dimensions. Clear an unknown result before solving backward, and supply enough independent measurements. Display rounding does not represent measurement uncertainty.
  • Calculations use 64 significant digits and full-precision π. Original reference drawings explain dimensions and are not drawn to the entered scale.

Common questions

Does slant and height determine which end radius is larger?

No. They determine the absolute radius difference. The retained slant-based inverse adds this difference to the known radius, selecting the larger end.

References

Require positive physical dimensions and real nondegenerate inverse roots while preserving documented limiting shapes. 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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