Triangle Calculator
Calculate all properties of any triangle
Enter Known Values
Enter any 3 values (sides or angles)
Triangle Diagram
About Triangle Calculations
A triangle is a polygon with three edges and three vertices. To fully define a triangle, you need at least three pieces of information, which can be any combination of sides and angles (with some restrictions).
Triangle Solution Methods
- SSS (Side-Side-Side): Three sides are known
- SAS (Side-Angle-Side): Two sides and the included angle
- ASA (Angle-Side-Angle): Two angles and the included side
- AAS (Angle-Angle-Side): Two angles and a non-included side
- SSA (Side-Side-Angle): Two sides and a non-included angle (may have 0, 1, or 2 solutions)
Key Formulas
- Sum of angles: A + B + C = 180°
- Law of cosines: c² = a² + b² - 2ab·cos(C)
- Law of sines: a/sin(A) = b/sin(B) = c/sin(C)
- Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Frequently Asked Questions
How many values do I need to solve a triangle?
You need at least 3 values to solve a triangle. This can be 3 sides (SSS), 2 sides and 1 angle (SAS or SSA), or 2 angles and 1 side (ASA or AAS). You cannot solve a triangle with 3 angles alone.
What is the triangle inequality theorem?
The triangle inequality states that the sum of any two sides must be greater than the third side. If this condition is not met, the three sides cannot form a valid triangle.
What is the law of cosines?
The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is c² = a² + b² - 2ab·cos(C), and is useful for solving SAS and SSS triangles.
What is the law of sines?
The law of sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. It is a/sin(A) = b/sin(B) = c/sin(C).
Can I solve a triangle with SSA?
Yes, but SSA (two sides and a non-included angle) can be ambiguous and may result in 0, 1, or 2 valid triangles. This is known as the ambiguous case.
How do I calculate the area of a triangle?
There are several methods: base × height ÷ 2, Heron's formula using three sides, or ½ab·sin(C) using two sides and the included angle.
What is Heron's formula?
Heron's formula calculates the area of a triangle when you know all three sides. First calculate s = (a+b+c)/2, then Area = √[s(s-a)(s-b)(s-c)].
Why must angles sum to 180 degrees?
In Euclidean geometry, the sum of interior angles in any triangle always equals 180 degrees (or π radians). This is a fundamental property of triangles in flat space.