Triangle Calculator
Solve a triangle from whatever you know — three sides, two sides and an angle, two angles and a side — and get every side, every angle, the area, perimeter, heights and the circle radii.
A triangle is fixed by three independent measurements, and which three you have decides the method.
How the triangle calculator works
A triangle is fixed by three independent measurements, and which three you have decides the method. Three sides (SSS) are solved by the law of cosines. Two sides with the angle between them (SAS) give the third side by the same law. Two angles and any side (ASA or AAS) are solved by the law of sines, since the third angle is whatever is left of 180°. Two sides and an angle that is not between them (SSA) is the ambiguous case: it can describe two different triangles, one, or none at all, and the calculator says which.
Area comes from ½ab·sin C once the parts are known, the heights from 2 × area ÷ each side, and the circumradius and inradius from the standard relations R = abc ÷ 4·area and r = area ÷ s.
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 |
|---|---|---|
| The 3-4-5 right triangle | 6 | area 6, right-angled |
| Two sides and the angle between (SAS) | 15.1554 | area 15.155 |
| Two angles and the side between (ASA) | 28.2629 | the third angle is 80° |
| An equilateral triangle of side 6 | 15.5885 | area 15.588 |
FAQFrequently asked questions
What do SSS, SAS and ASA mean?
They name which three measurements you have: three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA). Those three are enough to fix a triangle exactly.
Why is SSA called the ambiguous case?
Two sides and an angle that is not between them can describe two different triangles, one, or none. The calculator tells you which, and gives the second triangle when there is one.
Can any three numbers be a triangle?
No — any two sides must add up to more than the third. 1, 2 and 10 cannot close.
How is the area found from three sides?
Heron's formula: with s as half the perimeter, area = √(s(s−a)(s−b)(s−c)).
What are the circumradius and inradius?
The radii of the circle through all three corners and of the largest circle that fits inside. For a right triangle the circumradius is exactly half the hypotenuse.
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)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.