SSS Triangle Calculator — Solve a Triangle from Three Sides

About the SSS Triangle Calculator — Solve a Triangle from Three Sides

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.

When This Page Helps

Use this page when all three side lengths are known and you want the rest of the triangle without switching between formulas. It keeps the Law of Cosines, Heron's formula, and the derived heights, medians, and radii together so the solved triangle can be checked as a whole.

How to Use the Inputs

1. Enter side a, side b, and side c.
2. The calculator checks the triangle inequality first.
3. If valid, it computes all three angles via the Law of Cosines.
4. Area is calculated using Heron's formula.
5. Heights, medians, inradius, and circumradius are derived.
6. Click a preset to explore common triangles (3-4-5, equilateral, etc.).
7. Expand the reference table for triangle type formulas.

Example Calculation

Tips & Best Practices

• Always check the triangle inequality: each side must be less than the sum of the other two.
• If a²+b² = c², the triangle is right-angled at C.
• If a²+b² > c² for all arrangements, the triangle is acute.
• The inradius of a right triangle equals (a+b−c)/2.
• An equilateral triangle with side a has area = (√3/4)a².

When To Use This Calculator

Solve a triangle from three sides (SSS). Compute all angles, area, perimeter, heights, medians, inradius, and circumradius using Heron Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Frequently Asked Questions

• What is Heron's formula?

  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.

• What is the triangle inequality?

  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.

• How do I find angles from three sides?

  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.

• What is the difference between inradius and circumradius?

  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).

• Can two different triangles have the same three side lengths?

  No. By the SSS congruence theorem, three sides uniquely determine a triangle (up to reflection/rotation).

• What is a median of a triangle?

  A median connects a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect at the centroid.