THE NUMORIX GUIDE
How to use the Triangle Calculator
Last reviewed September 14, 2026
What this calculator does
The SSS mode uses Heron's formula and the law of cosines; SAS derives the missing side; ASA finds the third angle and uses the law of sines.
Formula and method
The SSS mode uses Heron's formula and the law of cosines; SAS derives the missing side; ASA finds the third angle and uses the law of sines. The result also reports perimeter, area, altitude to side a, all angles, and a basic triangle type.
Variables and inputs
Mode defaults to SSS with sides 3, 4, 5. SAS inputs are side a, included angle B in degrees, and side c. ASA inputs are angle A in degrees, side c, and angle B in degrees.
Worked example
For SSS 3, 4, 5: perimeter = 12, semiperimeter s = 6, area = sqrt(6*3*2*1) = 6, height to side a = 2*6/3 = 4, and angles are approximately A=36.87, B=53.13, C=90 degrees.
How to interpret the result
The three modes reconstruct a triangle from different independent measurements. The returned area and height use the same length unit as the side inputs.
Common mistakes to avoid
For SSS, verify the triangle inequality. In SAS and ASA, enter angles in degrees and make sure the angle sum remains below 180 degrees.
Assumptions and limitations
The SAS engine path currently passes the angle in radians directly into the subtractive term instead of applying cos(angle), so verify SAS results independently. Inputs are required to be positive even where a mathematical edge case might be discussed.
Practical use and checks
Choose the triangle mode that matches the measurements you actually have. With SSS inputs 3, 4, and 5, the calculator should return perimeter 12, area 6, and a right angle of 90 degrees. That is a useful check because the 3-4-5 relationship makes both Heron's formula and the Pythagorean theorem easy to verify. The SAS and ASA modes serve different information needs, so do not move a side or angle into a field merely because the label sounds similar. Use the area and perimeter outputs for material estimates, surveying, or a geometry check, and read the angle labels with their opposite side labels. Before trusting an SSS result, confirm the triangle inequality: each pair of sides must sum to more than the third. In angle modes, enter degrees as shown and make sure the supplied angles leave a positive third angle. A triangle that is nearly flat is especially sensitive to measurement rounding. The formulas describe ideal planar geometry and assume consistent length units; they do not account for thickness, curved edges, measurement uncertainty, or a physically impossible set of inputs. The existing SAS implementation deserves an independent check for precision-critical work, so compare it with a law-of-cosines calculation when the stakes are high.