Area histogram, percentile corridor, and invalid-sample gateLive current inputs
Math & Statistics
Triangle Distribution Calculator
Propagate independent side-length uncertainty through triangle validity, Heron area, perimeter, and angle calculations using a reproducible low-discrepancy sample instead of pretending measured sides are exact.
Mean valid-sample area-
Area standard deviation-
Area 5th percentile-
Area 95th percentile-
Invalid-triangle share-
Mean perimeter-
Mean angle A-
Valid deterministic samples-
MEASUREMENT UNCERTAINTY LANDSCAPE
Area histogram, percentile corridor, and invalid-sample gate
A deterministic normal-score lattice perturbs all three sides, filters impossible triangles, and displays the resulting area shape with its central 90% corridor.
PERCENTILE REGISTER
How the area and perimeter distributions move across probability levels
Percentiles are empirical values from the same reproducible sample used by the chart.
| Percentile | Area | Perimeter | Angle A | Triangle status |
|---|
MEASUREMENT ENTRY
Use one-sigma side uncertainties, not maximum tolerances
- Enter the best estimate of each side.
- Convert instrument tolerance to a standard uncertainty when needed.
- Keep all three side means and uncertainties in one length unit.
- Inspect the invalid share near a triangle-inequality boundary.
- Report percentile bounds with the assumed uncertainty model.
WHY A DISTRIBUTION IS NOT AN ERROR BAR
Heron's formula becomes skewed near geometric collapse
A symmetric side distribution does not imply a symmetric area distribution. Close to a+b=c, small side changes can eliminate the triangle or compress area toward zero.
The sample is deterministic so identical inputs reproduce identical rows and charts. It describes the entered independent-normal model, not every real measurement process.
NONLINEAR UNCERTAINTY PROPAGATION
Perturb sides first, then solve only geometrically valid samples
The deterministic sequence maps evenly spaced probabilities to normal scores. Each score triplet produces one side set; strict triangle inequalities are applied before Heron's formula or inverse cosine.
Detailed calculation process and general formulas
aᵢ = ā + sₐzᵢbᵢ = b̄ + sᵦzᵢ₊₁₂₁cᵢ = c̄ + s𝒸zᵢ₊₂₄₂Kᵢ = √[sᵢ(sᵢ-aᵢ)(sᵢ-bᵢ)(sᵢ-cᵢ)]Pq(K) = ordered valid-sample percentileSymbols, meanings, and units
- ā,b̄,c̄
- entered mean side lengthslength
- sₐ,sᵦ,s𝒸
- standard uncertainties of the sideslength
- zᵢ
- deterministic standard-normal scoredimensionless
- sᵢ
- sample semiperimeterlength
- Kᵢ
- area of one valid sampled trianglelength²
- Pq
- empirical qth percentilesame as quantity
GEOMETRY-RISK READOUT
Read spread, asymmetry, and feasibility separately
Three signals answer different questions about measured triangles.
Central spread
-The 5th-to-95th percentile corridor shows the middle 90% of valid sampled areas.
Feasibility loss
-Invalid samples quantify how often the perturbed sides violate a strict triangle inequality.
Angle migration
-Mean angle A reveals how the opposite-side uncertainty changes shape as well as size.
Decision takeaway: A narrow area band is not enough when a material share of side samples cannot form a triangle.
MODEL CHECKS
Questions to resolve before using the percentiles
- Are side errors independent?
- Are uncertainties one standard deviation?
- Are all sides in one unit?
- Is truncation at zero relevant?
- Is the triangle near degeneracy?
Applied decisions
Where uncertainty changes the geometric conclusion
Surveyed triangular parcel
Three tape or total-station lengths have small but nonzero standard uncertainties.
What the result clarifies: The area interval can support reporting precision without overstating exactness.
Nearly flat linkage
Two short members almost equal the longest member.
What the result clarifies: The invalid share warns that nominal validity is fragile under measurement error.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This is a deterministic uncertainty model based on independent normal side errors. Correlated observations, calibration bias, bounded tolerances, or a formal survey adjustment require a covariance-aware method. Invalid samples are excluded from area percentiles and reported separately.
Triangle Distribution Calculator | Side Uncertainty, Area Percentiles, and Validity FAQ
Why are some sampled triangles invalid?
Perturbed sides can violate a+b>c or another strict triangle inequality even when the mean sides are valid.
Why is the area distribution skewed?
Heron's formula is nonlinear and area is bounded below by zero.
Does the chart change on every reload?
No. The normal-score sequence is deterministic so the same inputs produce the same distribution.
Can I enter maximum tolerances?
Only after converting them to a defensible standard uncertainty for the assumed distribution.