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Math calculator

Cone Calculator

Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 16/16 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Right circular cone volume, slant, and surface.
Scope
Right circular cone
Verification
Engine tested · 16/16 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 3 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.1.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

VolumeV = (1/3)πr²h
Slantl = √(r² + h²)
iSame engine as /solid-geometry. The ISO 5167-5 cone meter is /calc/physics/iso-cone. Not a second engine.

How to use

1

Enter r and h

Both must be positive. r = 3, h = 4 is the 3-4-5 cone.

2

Read l, V, and A

3-4 cone has slant 5 and V = 12π.

Example calculations

Common configurations with formula and result.

ϟ

3-4-5 cone

r = 3, h = 4

l = 5, V = 12π
12π
ϟ

Unit cone

r = 1, h = 1

l = √2
π/3

Cone calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A right circular cone of radius r>0 and height h>0 has slant l=√(r²+h²), volume (1/3)πr²h, lateral area πrl, and total surface πr(r+l). This page is mode=cone on math.geometry.solid. Not ISO 5167-5. Not a frustum. Not CAS.
What it calculates
Right circular cone volume, slant, and surface.
Inputs
  • r
  • h
Outputs
  • volume
  • slant
  • lateral
  • surface
Formula
V = (1/3)πr²h, l = √(r²+h²)
Assumptions
  • Right circular cone
  • Not ISO 5167-5
  • Not a frustum
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
r=3 h=4 → l=5, V=12π
Validation cases

2 published on this page · 16/16 tests · Production surface contract 3/3 · View evidence

  • r=3 h=4 → l=5 V=12π
  • r=1 h=1 → V=π/3
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator, Ellipse Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

No. It is mode=cone on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the ISO 5167-5 cone meter?

No. That meter is /calc/physics/iso-cone. This page is a right circular solid.

Where does this run?

Locally in the browser by default. REST and MCP call the same solid-geometry engine.