Triangle Theorems Calculator
Solve any triangle by entering known sides and angles. Supports SSS, SAS, ASA, AAS, ASS, and AAA with full step-by-step output.
Triangle Solving Methods Summary
Triangle theorem selection depends on which values are known. This reference table shows each method, required inputs, and the solution approach.
Key Triangle Formulas
These triangle formulas are the foundation for all calculations in this tool.
Law of Sines
a / sin(A) = b / sin(B) = c / sin(C)
A = sin⁻¹( a · sin(B) / b )
Law of Cosines
a² = b² + c² - 2bc · cos(A)
cos(A) = (b² + c² - a²) / (2bc)
Triangle Area and Perimeter
s = (a + b + c) / 2 [semi-perimeter]
K = √(s · (s-a) · (s-b) · (s-c)) [Heron's formula]
r = √((s-a)(s-b)(s-c) / s) [inscribed circle radius]
R = (a · b · c) / (4K) [circumscribed circle radius]
The Ambiguous ASS Case
The ASS ambiguous case in triangles arises when one angle and two sides are known, but the angle is not between the two sides. Depending on the values, there may be zero, one, or two valid triangles.
- If angle A is 90 degrees or more and side a is less than or equal to side c, no triangle exists.
- If angle A is 90 degrees or more and side a is greater than side c, exactly one triangle exists.
- If angle A is less than 90 degrees and side a is greater than or equal to side c, one triangle exists.
- If angle A is less than 90 degrees and side a is less than side c, check sin(A) versus a/c: if sin(A) is less than a/c, two triangles exist; if equal, one triangle; if greater, no triangle.
Frequently Asked Questions
The SSS (Side-Side-Side) theorem states that when all three side lengths of a triangle are known, all three angles can be uniquely determined. Use the Law of Cosines formula cos A = (b² + c² – a²) / (2bc) to solve for each angle in turn.
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths opposite to angles A, B, C. It is used to solve AAS and ASA triangles and the ambiguous ASS case. Rearrange to find an unknown side or angle from known values.
The Law of Cosines states that a² = b² + c² – 2bc·cos(A). It generalizes the Pythagorean theorem to any triangle. Use it to solve SAS and SSS triangles. To find an angle from three known sides, rearrange to cos A = (b² + c² – a²) / (2bc) and apply the inverse cosine function.
Use Heron’s formula when all three sides are known. First calculate the semi-perimeter: s = (a + b + c) / 2. Then apply Area = √(s · (s-a) · (s-b) · (s-c)). This formula works for any triangle without needing to know any angles directly.
The ASS (Angle-Side-Side) case is ambiguous because the same inputs can sometimes produce two valid triangles, one triangle, or no triangle at all. This happens when the given angle is not the included angle between the two known sides. The calculator evaluates all conditions and displays both solutions when they exist.
AAA (Angle-Angle-Angle) only tells you the shape of a triangle and the ratio of its sides, not their actual lengths. Knowing all three angles does not uniquely define the triangle because an infinite number of similar triangles all share the same angle set. You can calculate the third angle from two given angles using the fact that angles sum to 180 degrees.
Yes, but only in the ASS (Angle-Side-Side) ambiguous case. When the given angle is acute, side a is shorter than side c, and sin(A) is less than a/c, two distinct triangles satisfy the given conditions. The calculator shows both complete solutions with all sides, angles, area, and perimeter for each valid triangle.