Side-constructed triangle versus angle-directed closureLive current inputs
Math & Statistics
Triangle Verification Calculator
Audit an independently measured triangle by comparing entered sides and angles against triangle inequality, angle sum, Law of Cosines, Law of Sines, and area consistency checks.
Overall verification status-
Entered angle sum-
Angle-sum error-
Largest cosine-derived angle error-
Law-of-Sines relative spread-
Area-method relative spread-
Triangle inequality-
Most restrictive check-
MEASUREMENT CLOSURE OVERLAY
Side-constructed triangle versus angle-directed closure
The blue polygon is constructed from the entered sides. The coral ray uses the entered A and b; its endpoint mismatch against the side-derived vertex exposes closure error spatially.
REDUNDANCY AUDIT
Five independent consistency tests with explicit tolerances
A pass means the entered redundancy agrees within tolerance; it does not certify measurement quality or instrument calibration.
| Check | Observed value | Reference value | Absolute or relative error | Tolerance | Status |
|---|
VERIFICATION SETUP
Enter measurements that were obtained independently
- Match each side with its opposite angle.
- Use one unit for all sides.
- Enter angles before forcing their sum to 180°.
- Set tolerances from instrument and workflow capability.
- Investigate the largest failed check rather than averaging errors away.
REDUNDANCY IS THE POINT
Six triangle measurements are not six independent degrees of freedom
A plane triangle has three independent dimensions. Extra side and angle measurements create checks that can expose transcription, labeling, or measurement problems.
Passing every test only means internal consistency at the chosen tolerance. Systematic bias can move all observations together and remain hidden.
TRIANGLE CLOSURE TEST
Use redundant measurements to reveal disagreement
Three sides already define a triangle. The entered angles are therefore redundant evidence that can be compared with side-derived angles and multiple area identities.
Detailed calculation process and general formulas
εsum = |A + B + C - 180°|A_SSS = cos⁻¹((b²+c²-a²)/(2bc))qA = a/sin A; qB = b/sin B; qC = c/sin CK_Heron = √(s(s-a)(s-b)(s-c))K_A = ½bc sin ASymbols, meanings, and units
- A,B,C
- independently entered anglesdegrees
- A_SSS,B_SSS,C_SSS
- angles derived from entered sidesdegrees
- qA,qB,qC
- Law-of-Sines diameter estimateslength
- εsum
- angle closure errordegrees
- K
- area computed by competing methodslength²
AUDIT INTERPRETATION
Different failures point to different likely causes
The register keeps them separate.
Angle closure
-A sum far from 180° points to angular measurement or transcription.
Opposite-pair mismatch
-Cosine and sine checks often expose swapped side-angle labels.
Area disagreement
-Heron versus SAS compares side-only and side-angle evidence.
Decision takeaway: Do not relax the tolerance merely to force a pass; revise the measurement process or document the discrepancy.
FOLLOW-UP ORDER
Investigate a failed triangle efficiently
- Confirm labels and vertex order
- Recheck unit conversion
- Inspect the largest angle residual
- Repeat the longest side
- Verify instrument calibration
- Document environmental and setup conditions
Applied decisions
Two very different sources of a failed audit
Swapped A and B labels
The angle sum remains exactly 180° but opposite-pair tests fail.
What the result clarifies: Closure alone is insufficient; cosine and sine redundancy reveals the swap.
Uniform angular bias
All three angles are shifted upward slightly.
What the result clarifies: Angle-sum error responds immediately even if their ordering remains plausible.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This tool checks internal Euclidean consistency against user-defined tolerances. It does not calibrate instruments, estimate correlated uncertainty, certify a survey closure, or replace domain-specific acceptance standards.
Triangle Verification Calculator | Side, Angle, and Area Consistency Audit FAQ
Can the angles sum to 180° and still be wrong?
Yes. They may not match the entered opposite sides.
Why use both cosine and sine checks?
They test different relationships and help localize labeling or measurement problems.
How should I choose tolerance?
Use instrument accuracy, procedure repeatability, unit conversion, and the consequence of error.
Does a pass prove the triangle is accurate?
No. It proves only internal agreement within the entered tolerances.