Triangle Calculator
A triangle has six measurements (three sides and three angles), and knowing the right three lets you find the rest. This calculator solves the three most common cases and adds the area, the perimeter and a description of the shape. It uses the law of cosines, the law of sines and Heron’s formula, and it warns you when the numbers cannot form a triangle.
How to use the triangle calculator
- Choose what you know: three sides (SSS), two sides and the angle between them (SAS), or two angles and the side between them (ASA).
- Enter the three values in order. Angles are in degrees.
- Read every side and angle, plus the area, the perimeter and the type of triangle.
Formula
- Law of cosines: c² = a² + b² − 2ab·cos C
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C
- Angles sum to 180°
- Heron’s formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a + b + c) ÷ 2
Worked examples
SSS: the 3-4-5 triangle
Sides 3, 4 and 5 give a right angle at C (90°), an area of 6 and a perimeter of 12.
SAS
Sides 3 and 4 with a 90° angle between them give c = √(9 + 16) = 5.
ASA
Angles 60° and 60° with a side of 10 between them give C = 60° and sides of 10 each: an equilateral triangle.
Which triangle facts you need
Three independent measurements pin a triangle down, as long as at least one is a side. The three cases below are the ones this calculator handles.
| Case | You know | First step |
|---|---|---|
| SSS | All three sides | Law of cosines for an angle |
| SAS | Two sides and the angle between | Law of cosines for the third side |
| ASA | Two angles and the side between | Third angle = 180° − A − B, then law of sines |
Cases that need care
Two angles and a side that is not between them (AAS) also works, since you can find the third angle first. Two sides and an angle that is not between them (SSA) can produce zero, one or two triangles, the “ambiguous case”, and is not covered here.
Classifying triangles
By angle: acute (all under 90°), right (one 90°) or obtuse (one over 90°). By sides: equilateral (all equal), isosceles (two equal) or scalene (none equal). The calculator reports both.
Real-world uses
Surveyors find distances they cannot measure directly, builders check roof pitches and navigators fix positions, all using these same laws.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Solve a triangle with sides 5, 12 and 13.
Show answer
Side a: 5 · Side b: 12 · Side c: 13
- Two sides of 7 and 10 meet at a 60° angle. Find the third side.
Show answer
Side a: 7 · Side b: 10 · Side c: 8.888194417
Common mistakes to avoid
- Entering sides that break the triangle inequality. Each side must be shorter than the other two combined.
- Putting the angle in the wrong position. In SAS the angle must be the one between the two sides.
- Forgetting that angles are in degrees, not radians.
Frequently asked questions
What is the triangle inequality?
The sum of any two sides must be greater than the third. Sides 1, 2 and 5 fail because 1 + 2 is less than 5.
How do I find the area without the height?
Use Heron’s formula with the three sides, or ½·a·b·sin C when you know two sides and the included angle.
What is the difference between SSS and SAS?
SSS gives all three sides. SAS gives two sides and the angle between them. Both fix the triangle’s shape completely.
Can I solve a right triangle here?
Yes, enter it as SSS or SAS. For a quick right-triangle answer see the Pythagorean theorem calculator.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.