Triangle Calculator
Solve any triangle from sides and angles — SSS, SAS, ASA, AAS configurations.
The Formula
Pythagorean: c² = a² + b²
Law of Cosines: c² = a² + b² − 2ab·cos(C)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
c = √(3² + 4²) = √(9+16) = √25
c = 5
Area = ½ × 3 × 4 = 6 sq units
How to Solve a Triangle
A triangle has three sides (a, b, c) and three angles (A, B, C) that always sum to 180°. You can find all six values from any of these input combinations: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), or SSA (two sides and a non-included angle — this one can have zero, one, or two solutions). Enter whatever values you know and the calculator solves for the rest.
The Key Laws for Solving Triangles
The Law of Sines states that each side divided by the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). Use it when you have two angle-side pairs. The Law of Cosines generalizes the Pythagorean theorem: c² = a² + b² - 2ab·cos(C). Use it for SSS and SAS cases. For right triangles, the Pythagorean theorem (a² + b² = c²) and basic trigonometry (sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent) are the most direct tools.
Finding Triangle Area
There are several ways to find the area depending on what you know. If you have base and height: Area = (1/2) x base x height. If you have two sides and the included angle: Area = (1/2) x a x b x sin(C). If you have all three sides, Heron's formula works: first compute the semi-perimeter s = (a + b + c)/2, then Area = √(s(s-a)(s-b)(s-c)).
Frequently Asked Questions
What is the Pythagorean theorem?
For any right triangle, the square of the hypotenuse (the side opposite the 90° angle) equals the sum of the squares of the other two sides: a² + b² = c². It is a special case of the Law of Cosines with angle C = 90° (since cos(90°) = 0, the 2ab·cos(C) term disappears). Common Pythagorean triples include 3-4-5, 5-12-13, and 8-15-17.
What is the SSA ambiguous case?
When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 valid triangles. If the given angle is obtuse, there is at most one solution. If the angle is acute, compare the side opposite the angle to the other given side: if it is shorter, there may be two solutions. If it equals the altitude from the third vertex, there is exactly one (a right triangle). This ambiguity is why SSA is the trickiest configuration.
How do I find the area of a triangle without the height?
Use Heron's formula if you know all three sides. Compute s = (a + b + c)/2, then Area = √(s(s-a)(s-b)(s-c)). Alternatively, if you know two sides and the included angle, use Area = (1/2) x a x b x sin(C). These formulas eliminate the need to find the height directly.
What are the different types of triangles?
By sides: equilateral (all three sides equal), isosceles (two sides equal), scalene (all sides different). By angles: acute (all angles less than 90°), right (one angle exactly 90°), obtuse (one angle greater than 90°). Every triangle has exactly one of each type from each category — a triangle cannot be both right and obtuse, for example.
What is the Law of Cosines used for?
The Law of Cosines (c² = a² + b² - 2ab·cos(C)) is used when you know all three sides (SSS) and want to find angles, or when you know two sides and the included angle (SAS) and want to find the third side. It reduces to the Pythagorean theorem when angle C = 90°. It is also used in navigation, surveying, and any field that requires calculating distances and angles in non-right triangles.