Free · Algebra and geometric relations

Cone Frustum Volume and Surface Measures

Relate two circular radii, perpendicular height, slant and surface areas of a conical frustum.

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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.

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Formula and method

For the worked example, Slant height (s) = 5.

V=πh(R²+Rr+r²)/3; l²=h²+(R−r)²

Worked example

Enter these known values and leave the other values blank.

Base radius (R)
5 cm
Top radius (r)
2 cm
Height (h)
4 cm
Slant height (s)
5 cm
Volume
163.36 cm³
Lateral surface area
109.96 cm²
Top surface area
12.566 cm²
Bottom surface area
78.54 cm²
Total surface area
201.06 cm²

Assumptions and limitations

  • The bases are parallel circles centered on a common perpendicular axis.
  • Use positive radii and perpendicular height. The volume and slant formulas are symmetric in the two radii; swapping the base labels preserves the geometry. Root inverses select a positive branch.
  • The tablespoon and teaspoon choices use metric volumes of 15 mL and 5 mL. The cup is a US customary cup.

Common questions

How do I use this relation?

Choose the relevant mode and enter independent known quantities. Leave a missing value empty where the control permits an inverse.

What do the displayed numbers mean?

The result follows the stated mathematical model. All dimensions must use consistent units; available unit selectors handle conversions.

References

Bookify replaces the captured rounded US gallon with the exact 3.785411784-liter definition. The US fluid ounce uses one 128th of the exact US liquid gallon, replacing the rounded captured reciprocal factor. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor. The US customary cup uses one sixteenth of the exact US liquid gallon, replacing the rounded captured cup volume.

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