AAA Triangle Calculator

What is an AAA (Angle-Angle-Angle) Triangle?

In Euclidean geometry, an AAA (Angle-Angle-Angle) triangle is a triangle where all three interior angles ($\alpha, \beta, \gamma$) are known. While knowing two angles allows you to instantly determine the third, AAA does not uniquely define the exact physical size of the triangle. Instead, it defines a family of infinitely many similar triangles with identical shapes and proportional side lengths.

Triangle Angle Sum Theorem

In planar Euclidean geometry, the sum of all three internal angles of any triangle always equals exactly $180^\circ$ ($\pi$ radians or $200$ gradians):

Therefore, given any two angles $\alpha$ and $\beta$, the missing third angle $\gamma$ is:

Proportions and the Law of Sines

According to the Law of Sines, the ratio of each side length to the sine of its opposite angle is constant:

Where $R$ is the circumradius. From this relationship, the relative ratio between sides $a$, $b$, and $c$ is:

Why AAA Does Not Prove Congruence

Unlike SAS (Side-Angle-Side), SSS (Side-Side-Side), or ASA (Angle-Side-Angle), AAA is a criterion for similarity, not congruence. An equilateral triangle with sides of length 1 cm has angles $60^\circ, 60^\circ, 60^\circ$, which is identical to an equilateral triangle with sides of length 100 meters. To calculate actual lengths, perimeters, or areas, at least one known side length is required as a scale factor.

Step-by-Step Example

Given angles $\alpha = 50^\circ$ and $\beta = 60^\circ$:

1. Calculate missing angle: $\gamma = 180^\circ - (50^\circ + 60^\circ) = 70^\circ$.
2. Compute sines: $\sin(50^\circ) \approx 0.7660$, $\sin(60^\circ) \approx 0.8660$, $\sin(70^\circ) \approx 0.9397$.
3. Normalize side ratios with $a = 1$: $b = \frac{0.8660}{0.7660} \approx 1.1305$, $c = \frac{0.9397}{0.7660} \approx 1.2267$.
4. Side ratio: $1 : 1.1305 : 1.2267$.

For related calculations and tools, see our 3 Sides Triangle Area Calculator, Half Square Triangle Calculator, and AAS Triangle Calculator.