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Math calculator

Triangle Solver

Solve a Euclidean triangle from SSS, SAS, ASA, AAS, or SSA (ambiguous case). Not the Pythagorean-only calculator. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 6/6 declared partitions (sss, sas, asa, aas, ssa, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Sides, angles, area, and an engine-linked Euclidean triangle figure (SSA draws both solutions).
Scope
Euclidean plane
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 3/3 property · digest 522b9356ef01
Legacy regression
10/10 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
6/6 golden · 2/2 CVP boundary · 6/6 invalid · 3/3 property · 2/2 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — law of sines and cosines
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 6/6 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Law of sinesa / sin A = b / sin B = c / sin C
Law of cosinesc² = a² + b² − 2ab cos C
Areaarea = (1/2) ab sin C
iAngles in degrees. Side a is opposite angle A. This is not the right-triangle Pythagorean page.

How to use

1

Choose the case

SSS needs three sides. SAS is sides a,b and included C. SSA is side a, opposite A, and side b.

2

Enter positive sides and angles in (0°, 180°)

A + B + C = 180°.

3

Read sides, angles, and area

SSA may list two solutions. The figure updates with the solved triangle.

Example calculations

Common configurations with formula and result.

ϟ

3-4-5

SSS right triangle

c² = 3² + 4²
C=90°, area=6
ϟ

SSA ambiguous

a=7, A=30°, b=10

two triangles
2 solutions

Triangle Solver specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A Euclidean triangle is determined by SSS, SAS, ASA, or AAS. SSA may yield zero, one, or two triangles (ambiguous case).
What it calculates
Sides, angles, area, and an engine-linked Euclidean triangle figure (SSA draws both solutions).
Inputs
  • mode
  • a?
  • b?
  • c?
  • A_deg?
  • B_deg?
  • C_deg?
Outputs
  • a
  • b
  • c
  • A_deg
  • B_deg
  • C_deg
  • area
  • solutions
Formula
law of sines / law of cosines
Assumptions
  • Euclidean plane
  • Not spherical trigonometry
  • Not Pythagorean-only
Units
  • length (consistent units)
  • degrees
Boundary conditions
  • triangle inequality fail → INVALID_INPUT
  • unknown mode (including hl) → INVALID_MODE
Example
mode=sss a=3 b=4 c=5 → C=90°
Validation cases

2 published on this page · 10/10 tests · Production surface contract 6/6 · View evidence

  • mode=sss a=3 b=4 c=5 → C_deg=90
  • mode=ssa a=7 A_deg=30 b=10 → solution_count=2
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for plane triangle sides and angles
  • CRC Standard Mathematical Tables — law of sines and cosines
    Supports: SSS/SAS/ASA/AAS law of cosines; SSA law of sines including the ambiguous case
Calculation version
1.0.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator . Explore all Geometry.

→ Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Arc Calculator Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. Not CAS. 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. › → 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. › → Hyperbola Calculator Standard hyperbola x²/a² − y²/b² = 1 (horizontal) or y²/a² − x²/b² = 1 (vertical). Same engine as Conic Section. Not CAS. Runs locally. › → Irregular Polygon Calculator Shoelace area of a simple 3–8 vertex polygon. Same engine as Plane Geometry. Not /regular-polygon. Not a triangle solver. Not CAS. Runs locally. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the Pythagorean calculator?

No. Right-triangle a²+b²=c² is /calc/math/pythagorean. This page solves general Euclidean triangles, including 3-4-5 as SSS.

What is the SSA ambiguous case?

Given a, A, and b, there can be two triangles when height < a < b and A is acute.

Where does this run?

Locally in the browser by default. REST and MCP call the same triangle-solver engine.