SSS Triangle Calculator
Know all three sides? Get every angle, the area, the perimeter and the heights — and a check that the three lengths can actually close into a triangle.
Three sides fix a triangle completely (that is the SSS congruence rule), so there is exactly one answer.
How the sss triangle calculator works
Three sides fix a triangle completely (that is the SSS congruence rule), so there is exactly one answer. Each angle comes from the law of cosines, cos A = (b² + c² − a²) ÷ 2bc, and the area from Heron's formula. The one thing to check first is the triangle inequality: any two sides must add up to more than the third, or the lengths cannot meet.
Formula: a² = b² + c² − 2bc·cos A; a / sin A = b / sin B = c / sin C; area = ½ab·sin C
Worked examples
| Inputs | Area | Note |
|---|---|---|
| 3, 4, 5 | 6 | the classic right triangle — area 6 |
| 7, 9, 12 | 31.305 | a scalene, obtuse triangle |
| 6, 6, 6 | 15.5885 | equilateral — every angle 60° |
FAQFrequently asked questions
What is the SSS rule?
Side-Side-Side: if you know all three sides, the triangle is completely determined — there is only one triangle with those lengths, up to reflection.
How do I find the angles from three sides?
The law of cosines rearranged: cos A = (b² + c² − a²) ÷ 2bc, then A is the inverse cosine of that.
Why do my sides not make a triangle?
Two of them must add up to more than the third. With 1, 1 and 5 the two short sides can never reach across.
How do I get the area from three sides?
Heron's formula, which this page uses: half the perimeter s, then area = √(s(s−a)(s−b)(s−c)).
Is a 3-4-5 triangle always right-angled?
Yes, and so is any multiple of it — 6-8-10, 9-12-15. They satisfy a² + b² = c² exactly.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.