Cone Calculator — Radius, Height, Slant, Surface Area, Volume

Enter any two values. Private by design — runs locally in your browser.

Diagram & Inputs

Tip: Enter any two values (r/d, h, l, S, L, or V). The calculator checks consistency if you provide more.

Results

How the Cone Calculator Works

A right circular cone is defined by radius r, height h, and slant height l with l = √(r² + h²). From these:

  • Lateral surface area: L = π r l
  • Total surface area: S = L + π r² = π r (l + r)
  • Volume: V = (1/3) π r² h

Provide any two values—such as r & h, r & S, h & V, l & S, or L & V. The tool solves for the missing dimensions (using closed-form rearrangements or a quick numeric solve when needed) and derives the rest. If you enter more than two values, it checks them for consistency.

Units: r, d, h, l are lengths; L and S use squared units; V uses cubed units. You can set decimal places and choose a π approximation for classroom alignment.

Formulas & Solvable Pairs (Quick Reference)

  • Given r & h: l = √(r²+h²), L=πrl, S=πr(l+r), V=(1/3)πr²h
  • Given r & l: h = √(l²−r²)
  • Given r & S: l = S/(πr) − r, then h = √(l²−r²)
  • Given r & L: l = L/(πr), then h = √(l²−r²)
  • Given r & V: h = 3V/(πr²)
  • Given h & V: r = √(3V/(πh))
  • Given l & S: r = (−l + √(l² + 4S/π))/2
  • Given L & S: r = √((S−L)/π), l = L/(πr)
  • Given l & V: solve (1/3)πr²√(l²−r²) = V for r (numeric)
  • Given S & V: solve πr(r + √(r² + (3V/(πr²))²)) = S (numeric)

5 Fun Facts about Cones

Unroll it, get a sector

Unwrap a cone’s side and you get a circular sector with radius l and arc length 2πr. That’s why L = π r l is “sector area” in disguise.

Net insight

Volume is exactly a third

A cone with the same base and height as a cylinder has 1/3 the volume. Archimedes proved it by “filling” a cylinder and sphere together.

Archimedes result

Slant sets everything

Knowing r and slant l fixes area, lateral area, and height in one go. It’s a Pythagorean shortcut often faster than starting with volume.

Quick solve

Angles change circle size

The cone’s apex angle controls how big the unwrapped sector is. A narrow cone makes a tiny “pizza slice”; a wide cone unwraps toward a full circle.

Apex intuition

Spot the frustum

Cut a cone with a plane parallel to its base and you get a frustum. Its volume: V = (1/3)πh(r₁² + r₁r₂ + r₂²)—useful for buckets and cups.

Real-world shape

Cone Calculator: FAQs

Which inputs are valid?

Any two of radius r (or diameter d), height h, slant height l, total surface area S, lateral area L, or volume V. More values are fine; the tool checks consistency.

What formulas are used?

l = √(r² + h²), L = π r l, S = L + π r², V = (1/3) π r² h. Closed forms are used where possible; otherwise a brief numerical solve is applied.

Does the calculator keep my data private?

Yes. Computation is entirely client-side; nothing is uploaded.

Can I change units, decimals, or π?

Yes. Choose a length unit, set decimal places, and pick a π approximation (native precision, 22/7, etc.).

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