How this calculator works
A cut parallel to a cone base leaves a frustum with radius changing linearly along its height. Integrating its circular slices gives the three-term volume. Slant mode first removes the radial change with the Pythagorean theorem. Radius reverse modes solve a quadratic and keep the nonnegative solution consistent with the larger-radius label.
Formula
V=πh(R²+Rr+r²)/3; l²=h²+(R-r)²; h=3V/[π(R²+Rr+r²)]. For q=3V/(πh), an unknown radius x satisfies x²+kx+k²=q.
Worked example
R=3 m, r=1 m and h=4 m give V=4π(9+3+1)/3=52π/3 m³. Equal radii of 2 m at height 3 m give the cylinder limit 12π m³. Setting r=0 gives a complete cone.
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Frequently asked questions
Can the top radius be zero?
Yes. This is the cone limit. The larger radius and vertical height must remain positive.
May the radii be equal?
Yes. The formula then reduces to cylinder volume and the slant equals the height.
Should I use exterior or interior radii?
Use interior measurements for capacity. Wall thickness and rounded edges are outside this ideal geometric model.