Solving a Triangle When All Three Sides Are Known
Three Sides Must First Make a Triangle
Three measured side lengths determine a triangle's area and angles, provided they can actually meet. This side-side-side case appears in layout, fabrication, surveying checks, truss sketches, and classroom geometry. Heron's formula finds area without first finding a height. The law of cosines then recovers each angle. Together they turn tape-measure data into a complete plane triangle and offer several ways to catch an impossible or mistyped dimension.
The triangle inequality is the first gate: any two sides must add to more than the third. If they add to exactly the third, the shape collapses into a line and has zero area. Once the shape is valid, the longest side faces the largest angle. Equal sides face equal angles. A right triangle has the familiar squared-side relationship, but Heron's formula works for acute and obtuse cases as well.
All three sides must use one unit and represent straight distances between the same three vertices. Measurement order only controls which angle receives label A, B, or C. Avoid rounded map-display lengths when accurate angles are required. A nearly flat triangle is highly sensitive to small side errors; dimensions that barely satisfy the triangle inequality can produce unstable area and angle estimates.
A 3-4-5 Geometry Check
The working equation is Area = sqrt(s(s-a)(s-b)(s-c)); angles follow from the law of cosines.
For sides 3, 4, and 5, semiperimeter is 6. Heron's formula gives the square root of 6×3×2×1, which is 6 square units. The law of cosines gives angles about 36.87°, 53.13°, and 90°. The perimeter is 12. Angle sum should be 180°, and the 5-unit side being opposite the 90° angle agrees with the Pythagorean check.
Model limit: Requires three positive side lengths satisfying the triangle inequality. Results describe a Euclidean plane triangle, not a curved-surface or surveyed geodesic triangle.
Near-Flat Shapes Magnify Measurement Error
Sides 3, 4, and 5 have semiperimeter 6. Heron's formula gives area sqrt(6×3×2×1) = 6 square units. Cosine calculations give angles 36.87°, 53.13°, and 90°, while perimeter is 12. The longest side faces the largest angle, and angle sum is 180°. Side values 1, 2, and 3 fail because the two shorter sides only equal the longest; they collapse to a line. Rejecting that case is part of correct calculation, not a missing output.
Now consider measured sides 10.00, 10.00, and 19.95 m. The triangle is technically valid but very flat, so centimetre-scale side error causes a large relative change in its small area and acute angles. It is poor geometry for locating a point. Repeat side measurements, use a labelled sketch, and seek an independent angle, coordinate, or diagonal check. For long survey lines, reduce slope distances and use an appropriate map projection or geodesic method. The plane solver is strongest for fabrication, classroom work, and local layouts where straight edges and Euclidean geometry are justified.
From Tape Measurements to a Defensible Layout
A common error is pairing an angle with the wrong opposite side in the law of cosines. Another is entering a diagonal or offset that does not connect the intended vertices. Heron's radicand can become slightly negative from extreme rounding near a degenerate shape. More importantly, ordinary plane formulas are not appropriate for long surveying distances on Earth's curved surface without projection or geodesic treatment.
Area uses squared versions of the side unit, while perimeter retains the original unit. The angles describe interior geometry. A very small angle opposite a short side can signal poor triangulation geometry because measurement noise is magnified. For construction layout, compare the calculated angles or diagonal with an independent measurement rather than using the same three sides to both create and “verify” the shape.
Mark three stable points, measure each side with consistent tension and endpoints, and repeat in reverse order. In fabrication, compare the predicted diagonal or angle with a square, template, or coordinate measurement. For land work, record slope distance versus horizontal distance and use an appropriate projection. Uncertainty can be explored by recalculating with each side at its plausible high and low value.
Keep a labeled sketch beside the side values so angle names remain unambiguous. Report units, measurement method, and whether the surface was treated as planar. The solver is useful for checking layouts and understanding geometry, but it cannot repair weak measurements or the wrong coordinate model. Validity, angle sum, longest-side ordering, and one independent field check form a practical review sequence. When the three vertices control costly work, propagate the tape or instrument tolerance through high and low side combinations before setting an angle or area acceptance limit. Avoid choosing vertex labels from memory after measurements are collected; transposed sides produce a valid triangle with angles assigned to the wrong corners.