Solve a triangle from three sides (SSS). Compute all angles, area, perimeter, heights, medians, inradius, and circumradius using Heron's formula and the Law of Cosines.
The SSS (Side-Side-Side) condition gives all three side lengths of a triangle. Given three positive lengths that satisfy the triangle inequality (each is less than the sum of the other two), there is exactly one triangle with those sides (up to congruence). This calculator solves the triangle completely.
The angles are found using the Law of Cosines: cos A = (b² + c² − a²) / (2bc), and similarly for B and C. The area comes from Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. From there, the three altitudes are hₐ = 2·Area/a, hᵦ = 2·Area/b, hᵧ = 2·Area/c. The medians follow from the formula mₐ = ½√(2b²+2c²−a²). The inradius (inscribed circle) is r = Area/s, and the circumradius (circumscribed circle) is R = a/(2 sin A).
SSS is one of the fundamental congruence conditions in Euclidean geometry, alongside SAS, ASA, AAS, and HL. It guarantees a unique triangle and avoids the ambiguous case that can arise with SSA. Engineers, architects, and surveyors frequently measure all three sides of a triangle in the field and use these formulas to compute angles and area.
This sss triangle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Side a, Side b, Side c, Decimal Places and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
cos A = (b² + c² − a²) / (2bc) Area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2 Height hₐ = 2·Area / a Median mₐ = ½√(2b²+2c²−a²) Inradius r = Area / s Circumradius R = a / (2 sin A)
Result: For a=3, b=4, c=5, the tool returns the solved sss triangle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in sss triangle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to sss triangle, with outputs tied directly to the form fields (Side a, Side b, Side c, Decimal Places). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
Heron's formula computes the area of a triangle from its three sides: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
For any valid triangle, each side must be strictly less than the sum of the other two: a < b+c, b < a+c, c < a+b. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
Use the Law of Cosines: cos A = (b²+c²−a²)/(2bc). Then A = arccos of that value. Repeat for B, and C = 180°−A−B.
The inradius (r) is the radius of the inscribed circle (tangent to all three sides). The circumradius (R) is the radius of the circumscribed circle (passing through all three vertices).
No. By the SSS congruence theorem, three sides uniquely determine a triangle (up to reflection/rotation).
A median connects a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect at the centroid.