What Is a Triangle Calculator?
A triangle calculator solves for the unknown sides, angles, area, perimeter, height, and other properties of a triangle when you know only some of them. Depending on which pieces of information you provide — three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA/AAS), or two sides and a non-included angle (SSA) — the calculator uses the appropriate formula to find the rest.
Triangle Basics
A triangle has three sides, three angles, and three vertices. The sum of its internal angles is always 180 degrees (Ï€ radians). The sum of any two sides is always greater than the third side (the triangle inequality). Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).
Key Formulas
Law of Sines
a / sin(A) = b / sin(B) = c / sin(C) = 2RWhere R is the radius of the circumscribed circle. Useful for ASA, AAS, and SSA cases.
Law of Cosines
c² = a² + b² − 2ab cos(C)Generalizes the Pythagorean theorem to any triangle. Used for SSS and SAS cases.
Area of a Triangle
Several formulas exist depending on what you know:
- Base and height: Area = ½ × base × height
- SAS: Area = ½ × a × b × sin(C)
- Heron's formula (SSS): Area = √( s(s-a)(s-b)(s-c) ) where s = (a+b+c)/2
- Two angles + side: Area = a² × sin(B) × sin(C) / (2 sin(A))
Common Congruence Cases
- SSS (side-side-side): all three sides known → unique triangle
- SAS (side-angle-side): two sides and included angle → unique triangle
- ASA (angle-side-angle): two angles and included side → unique triangle
- AAS (angle-angle-side): two angles and a non-included side → unique triangle
- SSA (side-side-angle): two sides and a non-included angle → may be 0, 1, or 2 valid triangles (the ambiguous case)
Example: SAS Triangle
Suppose you know sides a = 5 and b = 7, with included angle C = 60°.
- Find side c using the Law of Cosines: c² = 25 + 49 − 70 × cos(60°) = 74 − 35 = 39 → c ≈ 6.245
- Find angle A using the Law of Sines: sin(A)/5 = sin(60°)/6.245 → A ≈ 43.9°
- Find angle B: B = 180° − 60° − 43.9° = 76.1°
- Area via SAS: ½ × 5 × 7 × sin(60°) ≈ 15.16
Real-World Uses
Triangle calculations are essential in surveying, architecture, navigation, astronomy, physics, and engineering. Triangulation — determining a distance by measuring angles from two known points — is how GPS works, how ancient Greeks estimated the distance to the moon, and how modern surveyors map terrain. Roof trusses, bridges, and many structures rely on triangles for strength and stability.
Frequently Asked Questions
What is the ambiguous case (SSA)?
When you know two sides and a non-included angle, there may be zero, one, or two valid triangles. This happens because the sine of an angle and the sine of its supplement are equal — you cannot tell from the sine alone which angle is the correct one.
How do I know if three sides form a valid triangle?
Check the triangle inequality: the sum of any two sides must be greater than the third. If the sum of the two shorter sides is less than or equal to the longest side, you cannot form a triangle.
What is Heron's formula?
Heron's formula gives the area from three sides alone: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2. It is named after Hero of Alexandria.