What this calculator does
A triangle is fully determined by three independent measurements, but which three you actually have varies: sometimes it is all three sides, sometimes two sides and the angle between them, and sometimes two angles and a side. This triangle calculator covers all three, so it works as a single triangle formula lookup whichever combination you start with.
Switching the known-values option changes which method runs underneath, from Heron’s formula through to the law of cosines and the law of sines, but the area is always shown as the headline result, with the remaining sides and angles given alongside it.
The formula
With three sides known, Heron’s formula gives the area directly, and the law of cosines gives each angle. With two sides and the included angle, the third side comes from c² = a² + b² − 2ab·cos C, then the angles and area follow. With two angles and the included side, the third angle is 180° minus the other two, and the law of sines, a/sin A = b/sin B = c/sin C, gives the remaining sides.
| Term | Meaning |
|---|---|
| SSS | Three known sides, solved with Heron’s formula and the law of cosines. |
| SAS | Two sides and the angle between them, solved with the law of cosines. |
| ASA | Two angles and the side between them, solved with the law of sines. |
The inputs explained
| Field | What to enter |
|---|---|
| Known values | Choose which measurements you know: three sides, two sides and the angle between them, or two angles and the side between them. |
| Side a (SSS/SAS), or Angle A in degrees (ASA) | A side length (SSS or SAS), or Angle A in degrees (ASA). |
| Side b (SSS/SAS), or Angle B in degrees (ASA) | A side length (SSS or SAS), or Angle B in degrees (ASA). |
| Side c (SSS), included angle C in degrees (SAS), or included side c (ASA) | The third side (SSS), the included angle in degrees (SAS), or the included side (ASA). |
When to use it
A triangle calc from three measured sides
A surveyed plot, panel or truss given as three side lengths is solved with the SSS method here, returning area, perimeter and all three angles at once.
A triangle formula from two sides and an angle
Navigation, bearing and trigonometry problems often give two distances and the angle between them, exactly the SAS case, without a third side measured directly.
A triangle solved from two angles and a baseline
Surveying by triangulation typically fixes two angles from a known baseline length, the ASA case, and works out the remaining unknown distances from there.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Triangle calculator formula: solving from three sides
Two sides fixed at 8 and 9, with the third side varying across the range a valid triangle allows, using the three-sides method.
| Side a | Area | Angles (opposite a, b, c) |
|---|---|---|
| 3 | 11.832 | 19.19° · 61.22° · 99.59° |
| 5 | 19.900 | 33.56° · 62.18° · 84.26° |
| 7 | 26.833 | 48.19° · 58.41° · 73.40° |
| 9 | 32.249 | 63.61° · 52.78° · 63.61° |
| 11 | 35.496 | 80.41° · 45.82° · 53.78° |
| 16 | 22.933 | 140.43° · 18.57° · 21.00° |
Triangle formula: solving from two sides and the angle between them
Two sides fixed at 8 and 6, with the included angle between them varying, using the two-sides-and-angle method.
| Included angle | Area | Third side |
|---|---|---|
| 30° | 12.000 | 4.106 |
| 60° | 20.785 | 7.211 |
| 90° | 24.000 | 10.000 |
| 120° | 20.785 | 12.166 |
| 150° | 12.000 | 13.533 |
Questions
How do you use this triangle calculator?
Choose which three measurements you already know, three sides, two sides and the angle between them, or two angles and the side between them, then enter those values. The calculator picks the matching method automatically.
What is the triangle formula for area from three sides?
Heron’s formula: find the half-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). It needs no separate height measurement.
What if I only know two angles and no sides at all?
Two angles alone fix the triangle’s shape but not its size, since it could be scaled up or down indefinitely. At least one side length is needed alongside the angles to pin down actual measurements.
Does this triangle calc work for a right-angled triangle?
Yes, a right angle is just a 90° value entered like any other angle, or a triangle where the three sides happen to satisfy Pythagoras. For a right triangle specifically, the dedicated right triangle calculator may be quicker.
For a right-angled triangle from its two legs, see the right triangle / Pythagoras calculator. For area alone from base and height, three sides, or two sides and an angle, see the triangle area calculator.