Example Data Table
These examples show typical inputs and the computed missing side.
| Mode | Inputs | Key output |
|---|---|---|
| SAS | a=5, b=7, C=60° | c≈6.245 |
| AAS/ASA | A=40°, B=75°, known a=12 | b≈17.265, c≈15.338 |
| Right | a=3, b=4 | c=5 |
| SSS | a=6, b=8, c=10 | Right triangle, area=24 |
Numbers are rounded for display. Your results depend on chosen decimals.
Formulas Used
- Cosine law (SAS/SSS): c = √(a² + b² − 2ab cos C) and cyclic variants for angles.
- Sine law (AAS/ASA): a/sinA = b/sinB = c/sinC, with C = 180° − A − B.
- Pythagoras (right triangles): c² = a² + b², or a = √(c² − b²).
- Heron’s formula (area): s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)).
- Heights: ha = 2·Area/a (and similarly for b, c).
How to Use This Calculator
- Select a calculation mode that matches your known values.
- Enter required sides and angles using consistent units.
- Choose decimal places and optionally show formula notes.
- Press Calculate to view results above the form.
- Use the download buttons to export CSV or PDF.
FAQs
1) What does “included angle” mean in SAS?
It is the angle between the two known sides. If you know two sides and the angle between them, the cosine law finds the third side reliably.
2) Why must angles sum to less than 180°?
A triangle’s interior angles always total 180°. If two angles already reach 180° or more, the third angle cannot be positive, so no triangle exists.
3) How do you check if three sides make a valid triangle?
All sides must be positive, and each pair must sum to more than the remaining side. This triangle inequality prevents flat or impossible shapes.
4) Can I solve any triangle with AAS/ASA mode?
Yes, if you provide two angles and one side. The missing angle is computed from 180°, then the sine law scales the remaining sides.
5) Why are my outputs slightly different after changing decimals?
Rounding changes how many digits are shown. Internally, calculations use full floating precision, then the displayed values are formatted to your chosen decimals.
6) Does the right triangle mode assume a specific angle?
It assumes one angle is exactly 90°. Enter the two known sides based on the unknown you choose, and the tool uses the Pythagorean relationship.
7) What unit should I use for angles?
Angles must be entered in degrees. Side units can be any length unit, but keep them consistent so area and perimeter remain meaningful.
8) How do the CSV and PDF exports work?
After a successful calculation, the result is saved for your session. The export buttons download the latest computed values as a CSV row or a one-page PDF.