SAS Triangle Calculator
Two sides with the angle between them fix a triangle exactly. Enter them to get the third side, both remaining angles and the area.
The included angle is the key: with two sides and the angle between them there is exactly one triangle, so no ambiguity.
How the sas triangle calculator works
The included angle is the key: with two sides and the angle between them there is exactly one triangle, so no ambiguity. The third side comes from the law of cosines, c² = a² + b² − 2ab·cos C, and the area directly from ½ab·sin C without needing the third side at all.
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 |
|---|---|---|
| Sides 5 and 7 with 60° between | 15.1554 | area 15.155 |
| Sides 8 and 6 with 90° between | 24 | a right triangle — area 24 |
| A shallow 20° angle | 20.5212 | area 20.52 |
FAQFrequently asked questions
What does SAS mean?
Side-Angle-Side: two sides and the angle between them. That combination fixes a triangle exactly, unlike two sides and an angle somewhere else.
Why is SAS never ambiguous?
Because the angle is locked between the two known sides, so the third side has only one possible length.
Can I find the area without the third side?
Yes — area = ½ab·sin C uses only the two sides and the included angle.
What if my angle is not between the sides?
Then it is the SSA case, which can have two answers; use the SSA page instead.
Does the angle have to be less than 180°?
Yes. At 180° the two sides lie flat in a line and there is no triangle.
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)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.