AAS Triangle Calculator

Understanding the AAS (Angle-Angle-Side) Triangle Theorem

In geometry, the AAS (Angle-Angle-Side) condition occurs when you know the measures of two consecutive interior angles of a triangle and the length of a non-included side (a side that is opposite one of the given angles). Because the three interior angles of any Euclidean triangle always sum to $180^\circ$ ($\pi$ radians), knowing two angles immediately determines the third angle. Once all three angles and one side are known, the triangle is uniquely determined up to congruence by the Law of Sines.

How to Solve an AAS Triangle Step-by-Step

Solving an AAS triangle involves three fundamental geometric and trigonometric steps:

1. Find the Third Angle ($\gamma$): Using the triangle angle sum theorem: $$\gamma = 180^\circ - (\alpha + \beta)$$
2. Apply the Law of Sines: The Law of Sines states that the ratio of any side length to the sine of its opposite angle is constant: $$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}$$ Solving for the unknown side lengths $b$ and $c$ when side $a$ is given: $$b = \frac{a \cdot \sin\beta}{\sin\alpha}, \quad c = \frac{a \cdot \sin\gamma}{\sin\alpha}$$
3. Compute Area and Perimeter: The perimeter is simply $P = a + b + c$. The area can be determined directly using the sine area formula: $$\text{Area} = \frac{1}{2} a b \sin\gamma = \frac{a^2 \sin\beta \sin\gamma}{2 \sin\alpha}$$

AAS vs. ASA Triangles: What Is the Difference?

Both AAS and ASA triangles share the exact same known inputs: two angles and one side length. The difference lies in the position of the known side:

• AAS (Angle-Angle-Side): The known side is non-included; it sits opposite one of the given angles.
• ASA (Angle-Side-Angle): The known side is included between the two given angles.

Because knowing two angles always yields the third angle, any AAS triangle problem can be converted directly into an ASA problem, and both provide sufficient information to prove geometric congruence and solve all dimensions uniquely. For triangles where only angles are known without any side lengths, explore our AAA Triangle Calculator.

Key Geometric Properties Calculated

Beyond finding missing side lengths and angles, this solver calculates:

• Altitudes ($h_a, h_b, h_c$): The perpendicular heights from each vertex to the opposite side, computed via $h_a = \frac{2 \cdot \text{Area}}{a} = b \sin\gamma$.
• Inradius ($r$): The radius of the largest circle inscribed inside the triangle: $r = \frac{\text{Area}}{s}$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter.
• Circumradius ($R$): The radius of the circle passing through all three vertices: $R = \frac{a}{2\sin\alpha}$.