Frustum Calculator
Right circular frustum V = (1/3)πh(R²+Rr+r²) from two radii and height. Same engine as Solid Geometry. Not /cone. Not ISO 5167-5. Not CAS. Runs locally.
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 frustum volume, slant, and surface from two radii and height.
- Scope
- Right circular frustum
- 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
- 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- Embedded snapshot: unpublished · Build schema 1.1.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.
How to use
Enter R, r, and h
Both radii and the height must be positive. Default R=2, r=1, h=3. Equal radii r=R are a valid cylinder.
Read V, l, and A
2-1-3 has volume 7π. Equal radii 1 and 1 with h=1 match the unit cylinder.
Example calculations
Common configurations with formula and result.
2-1-3
R = 2, r = 1, h = 3
Cylinder identity
R = 1, r = 1, h = 1
Frustum calculator specification
Version 1.1.0 · Engine tested
- Engine tested 16/16 tests · Production surface contract 3/3
- Named expert review Not performed
- Calculation version 1.1.0
- Definition
- A right circular frustum of bases R>0, r>0 and height h>0 has volume (1/3)πh(R²+Rr+r²) and slant √(h²+(R−r)²). r=R recovers a cylinder. This page is mode=frustum on math.geometry.solid. Not /cone. Not ISO 5167-5. Not CAS.
- What it calculates
- Right circular frustum volume, slant, and surface from two radii and height.
- Inputs
- R
- r
- h
- Outputs
- volume
- slant
- lateral
- surface
- Formula
V = (1/3)πh(R²+Rr+r²), l = √(h²+(R−r)²)- Assumptions
- Right circular frustum
- r=R is a valid cylinder
- Not /cone
- Not ISO 5167-5
- Not prism / pyramid
- Not CAS
- Units
- length (consistent units)
- volume (length³)
- area (length²)
- Boundary conditions
- R ≤ 0 or r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
- unknown mode → INVALID_MODE
- Example
- R=2 r=1 h=3 → V=7π
- Validation cases
3 published on this page · 16/16 tests · Production surface contract 3/3 · View evidence
- R=2 r=1 h=3 → V=7π
- R=1 r=1 h=1 → V=π cylinder=true
- R=2 r=1 h=0 → VALUE_MUST_BE_POSITIVE
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.1.0
Related tools
Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator . Explore all Geometry.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second solid engine?
No. It is mode=frustum on math.geometry.solid, the same engine as /calc/math/solid-geometry.
Is this the cone page?
No. The right circular cone stays on /calc/math/cone. Equal radii here recover a cylinder, not a cone.
Is this the ISO 5167-5 cone meter?
No. That meter is /calc/physics/iso-cone. This page is a Euclidean solid of revolution.
Where does this run?
Locally in the browser by default. REST and MCP call the same solid-geometry engine.