Source of Data & Methodology

Authoritative source: National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, Geometry of 3D shapes; and CRC Standard Mathematical Tables & Formulae (selected identities). NIST DLMF (accessed Oct 25, 2025).
All calculations strictly follow the formulas and definitions from this source.

The formulas explained

For a right circular cone with base radius \(r\), height \(h\), and slant height \(l=\sqrt{r^2+h^2}\):

Base area: \(\displaystyle A_B=\pi r^2\)
Lateral area: \(\displaystyle A_L=\pi r\,l\)
Total area: \(\displaystyle A_T=\pi r(r+l)\)
Volume: \(\displaystyle V=\frac{1}{3}\pi r^2 h\)
Apex angle (full): \(\displaystyle \Theta = 2\,\arctan\!\left(\frac{r}{h}\right)\),   half-angle \(\displaystyle \alpha=\arctan\!\left(\frac{r}{h}\right)\).

Glossary of variables

  • r: base radius (length)
  • d: base diameter \(d=2r\) (length)
  • h: height (axis) (length)
  • l: slant height (generator) (length)
  • A_B: base area (\( \text{length}^2 \))
  • A_L: lateral surface area (\( \text{length}^2 \))
  • A_T: total surface area (\( \text{length}^2 \))
  • V: volume (\( \text{length}^3 \))
  • \(\Theta\): full apex (aperture) angle in degrees

How it works: a step-by-step example

Suppose you have a cone with r = 5 cm and h = 12 cm.

  1. Compute \(l=\sqrt{r^2+h^2}=\sqrt{5^2+12^2}=13\text{ cm}\).
  2. Base area \(A_B=\pi r^2=\pi\cdot 25\approx 78.540\text{ cm}^2\).
  3. Lateral area \(A_L=\pi r l=\pi\cdot 5\cdot 13\approx 204.204\text{ cm}^2\).
  4. Total area \(A_T=\pi r(r+l)=\pi\cdot 5\cdot 18\approx 282.743\text{ cm}^2\).
  5. Volume \(V=\tfrac13\pi r^2 h=\tfrac13\pi\cdot 25\cdot 12\approx 314.159\text{ cm}^3\).
  6. Apex angle \(\Theta=2\arctan(r/h)\approx 2\arctan(5/12)\approx 45.238^\circ\).

Frequently asked questions

Which input pair should I use?

Use the pair you actually know. For metal layout, r+l is common; for classroom problems, r+h or d+h are typical; for process design, V+h sometimes appears.

Is this valid for oblique cones?

No. These closed forms assume a right circular cone. Oblique cones need different methods.

What is the “apex angle” vs “half-angle”?

The full apex angle is at the tip across the axis; the half-angle is measured from the axis to the generator.

Can I switch units?

Yes. Pick Metric or Imperial and select a display length unit. The tool keeps results coherent.

Do rounding and display affect accuracy?

The engine computes in double precision and rounds for display. You can copy raw values from the browser console if needed.

Tool developed by Ugo Candido. Content verified by the CalcDomain Editorial Board.
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