How this calculator works
A cut parallel to a cone’s base leaves two concentric circular ends. The slant, vertical height and difference of radii form a right triangle. The curved area uses the sum of radii and the slant. Adding the larger circular cap gives the open-top surface; adding both caps gives the closed total. Radius reverse modes solve the resulting quadratic with a known slant. Every solution is checked for positive vertical height. Equal radii describe the cylindrical limit; a zero top radius describes a complete cone.
Formula
s² = h²+(R-r)²; curved area L = π(R+r)s; base = πR²; top = πr²; closed total S = π[(R+r)s+R²+r²]. For unknown radius u and known radius v: u²+su+v²+sv-S/π=0.
Worked example
R=5 m, r=2 m and h=4 m give s=5 m. The curved wall is 35π m², the larger cap is 25π m² and the smaller cap is 4π m². The open-top total is 60π m² and the closed total is 64π m².
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Frequently asked questions
Does total include the hole at the top?
The closed total includes a circular top cap. For an open vessel, use the separate open-top result, which includes only the larger base and curved wall.
Can I enter a cylinder?
Yes. Equal radii make the radius difference zero, so the slant equals the vertical height and the curved area becomes 2πRh.
Does this allow an angled cut?
No. The cut must be parallel to the base and the cone must be right circular. An oblique cut produces a different boundary.