Triangle Solver (Heron’s Area, Law of Cosines & Inradius)

Solve SSS triangles for area, all three interior angles, inradius, and circumradius.

Triangle Area (Heron’s Formula)
26.833 sq units
Semiperimeter (s) 12 units
Inscribed Circle Radius (r = A/s) 2.236 units
Circumcircle Radius (R = abc/4A) 4.696 units
Opposite Angle C 73.399 deg
Inputs
Triangle side length a.
Triangle side length b.
Triangle side length c.
Results & Graph
Triangle Area (Heron’s Formula)
26.833 sq units
Semiperimeter (s)
12 units
Inscribed Circle Radius (r = A/s)
2.236 units
Circumcircle Radius (R = abc/4A)
4.696 units
Opposite Angle C
73.399 deg

Triangle Area (Heron’s Formula)

Graph 1 • Side A Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 sq units 18.9 sq units 37.7 sq units 50%: 13.95 sq units 13.9 50% 75%: 20.833 sq units 20.8 75% 100%: 26.833 sq units 26.8 100% 125%: 31.674 sq units 31.7 125% 150%: 34.936 sq units 34.9 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Triangle Area (Heron’s Formula) 26.833 sq units Semiperimeter (s) 12 units Inscribed Circle Radius (r = A/s) 2.236 units Circumcircle Radius (R = abc/4A) 4.696 units
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Triangle Area (Heron’s Formula)26.833 sq unitsSemiperimeter (s)12 unitsInscribed Circle Radius (r = A/s)2.236 unitsCircumcircle Radius (R = abc/4A)4.696 unitsOpposite Angle C73.399 deg
Summary: Sides (7, 8, 9) have semiperimeter s = 12, area = 26.8328, and angle C = 73.4°.
Summary: Sides (7, 8, 9) have semiperimeter s = 12, area = 26.8328, and angle C = 73.4°.

Formula & Mathematical Methodology: Triangle Solver Mathematical Identity

s = (a+b+c)/2
Area = s(s−a)(s−b)(s−c)
cos(C) = a²+b²−c²2ab
s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]; cos(C) = (a²+b²−c²)/(2ab)
  • sideA: Side A (numeric parameter)
  • sideB: Side B (numeric parameter)
  • sideC: Side C (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Triangle Solver

Solve SSS triangles for area, all three interior angles, inradius, and circumradius.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]; cos(C) = (a²+b²−c²)/(2ab).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Triangle Solver via s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]; cos(C) = (a²+b²−c²)/(2ab).

Frequently Asked Questions

What mathematical formula does the Triangle Solver (Heron’s Area, Law of Cosines & Inradius) use?

It directly evaluates s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]; cos(C) = (a²+b²−c²)/(2ab) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Geometry & Trigonometry Guide

How to Use the Triangle Solver (Heron’s Area, Law of Cosines & Inradius): Practical Guide & Visual Analysis

Calculate any triangle’s area using Heron’s formula, interior angles via the Law of Cosines, semiperimeter, inradius r, and circumradius R. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching triangle solver (heron’s area, law of cosines & inradius), triangle solver, triangle heron law of cosines, geometry & trigonometry calculator, this interactive tool helps you calculate any triangle’s area using Heron’s formula, interior angles via the Law of Cosines, semiperimeter, inradius r, and circumradius R.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Side A, Side B, Side C, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Triangle Solver), Evaluates Triangle Solver via s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]; cos(C) = (a²+b²−c²)/(2ab). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).