Right Angle Compatibility for a Chosen Hypotenuse

Formula and method

Sides 3, 4 and candidate hypotenuse 5 are compatible with a right angle opposite 5.

Three sides: |a²+b²−c²|/(a²+b²+c²)≤10⁻⁶⁰. Two angles: |α+β−π/2|≤10⁻⁶⁰radians. Side/angle ratios use absolute error<0.005.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• c is the candidate hypotenuse. A negative verdict concerns the right angle opposite that chosen side; for an unlabeled triangle put the longest side in c. All positive side lengths must form a nondegenerate triangle.
• All captured modes, six side/angle choices, editable controls, length units and nine angle units remain available. Hidden residual inverses retain their captured algebraic branch; they are not a list of every solution.
• The three-side verdict uses a relative squared-side residual at most 1e−60, independent of the common length unit. The two-angle verdict uses a 1e−60 radian sum residual. These reflect numerical compatibility at the working 64 digit precision, not measurement uncertainty or proof of an exact right angle.
• The side/angle mode preserves the captured absolute dimensionless ratio error strictly below0.005. That is neither an angle tolerance nor a claim of precision. An SSA-compatible right triangle need not be the only triangle admitted by those data.
• Rounded measurements may fail the strict side/angle-sum numerical tests. Near a degenerate triangle, or with an acute angle numerically indistinguishable from 0° or 90°, the arithmetic may not resolve a valid bound; an error means insufficient numerical separation rather than a new physical restriction.
• In the side/angle modes, ratios use the angle you actually entered through exact complementary-angle identities. A hidden complementary angle may round to an endpoint at finite precision; its rounded value is not used to discard information from the entered angle.

Common questions

References

• Euclid: converse of the Pythagorean theorem
• OpenStax: right-triangle trigonometry
• Calculation definition and unit reference

Preserve all six captured equations and their inverse branches. Replace the unit-dependent absolute-zero side verdict with an explicitly disclosed dimensionless relative squared-side residual limit1e-60, derived from the existing arithmetic absolute tolerance. Preserve the angular1e-60radian resolution and the captured strictly-less-than0.005 absolute ratio tolerance, using exact sign comparisons for verdict boundaries. These are numerical compatibility statements, not exact geometric certification or measurement uncertainty. The selected c remains the candidate hypotenuse; do not silently sort sides or assert unique SSA geometry. Evaluate direct trigonometric side/ratio variants only for the captured acute-angle domain. Apply that same lazy guard to their three forward consistency identities, so a supported-to-invalid-angle transition cannot evaluate enormous trigonometric arguments through retained prior outputs. Inverse angle branches do not require an already-known angle. In the six side/angle modes, evaluate each ratio from the angle actually entered: tan(alpha)=cot(beta), sin(alpha)=cos(beta), cos(alpha)=sin(beta). The beta-input modes use those cofunction identities directly instead of subtracting a tiny beta from pi/2 and then taking trigonometry of the rounded complement. Alpha-input modes retain their original expressions. This preserves tiny-angle information at unchanged precision; hidden complement controls and every captured inverse-angle branch remain unchanged. Retain all positive, acute-angle and nondegenerate-triangle domains using sign-based comparisons, avoiding accidental treatment of tiny positive quantities as zero. Apply the same positive-length domain to the three captured side/angle controls, which lacked it. Numerical resolution near a degenerate triangle is disclosed; no physical magnitude cap is introduced.

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