Free AAS Triangle Calculator

AAS Triangle Calculator: Online Angle-Angle-Side Solver

The AAS Triangle Calculator is a free online tool designed to solve any triangle when two consecutive angles and a non‑included side are known. This angle angle side calculator instantly provides the missing angle, the two remaining sides, the triangle’s height, and its area. Serving as both a triangle congruence calculator and a law of sines calculator, it automates all trigonometric steps so you can focus on interpreting the results.

AAS Triangle Congruence

In geometry, two triangles are congruent if they have exactly the same size and shape. The AAS (Angle‑Angle‑Side) congruence criterion states that if two angles and the side opposite one of them in one triangle equal the corresponding two angles and side of another triangle, the triangles are congruent. This holds because once two angles are fixed, the third angle is determined by the sum of interior angles ($\pi$ or $180^\circ$), and the law of sines forces the side lengths to match.

An AAS triangle is defined by the following knowns:

• One side $a$,
• The angle $\alpha$ opposite that side,
• The angle $\beta$ adjacent to that side.

The side is not included between the two angles—this is what distinguishes AAS from ASA (Angle‑Side‑Angle). The uniqueness of the AAS configuration ensures that given valid input (angles summing to less than $180^\circ$), there is exactly one triangle that fits the description.

Solving an AAS Triangle with the Law of Sines

The law of sines is the fundamental relationship used to solve AAS triangles. It states:

$$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} = 2R,$$

where $R$ is the circumradius of the triangle. This constant ratio allows every unknown side to be expressed in terms of the known side and the sines of the angles.

Step 1: Find the Missing Angle

Since the angles in a triangle always sum to $180^\circ$ (or $\pi$ rad), the third angle $\gamma$ is:

$$\gamma = 180^\circ - \alpha - \beta \quad (\text{or } \gamma = \pi - \alpha - \beta).$$

Step 2: Determine the Two Unknown Sides

Using the law of sines with side $a$ and the three angles:

$$b = a \cdot \frac{\sin\beta}{\sin\alpha}, \qquad c = a \cdot \frac{\sin\gamma}{\sin\alpha}.$$

Equivalently, one can first compute the constant $k = a / \sin\alpha$ and then multiply by $\sin\beta$ and $\sin\gamma$ to obtain $b$ and $c$. These calculations are the essence of what any law of sines calculator performs.

AAS Triangle Area Formula

The area of an AAS triangle can be expressed directly in terms of the known side and angles. One approach uses the base‑height formula. Let side $b$ be the base; the height $h$ relative to that base is:

$$h = a \cdot \sin\gamma,$$

because the altitude from the vertex opposite side $a$ creates a right triangle with hypotenuse $a$ and angle $\gamma$. Substituting $b$ and $h$ into $\text{Area} = \frac12\,b\,h$ gives:

$$\text{Area} = \frac12 \left(a \frac{\sin\beta}{\sin\alpha}\right) \left(a \sin\gamma\right) = \frac12 a^{2} \frac{\sin\beta \,\sin\gamma}{\sin\alpha}.$$

Using $\gamma = \pi - \alpha - \beta$ and the identity $\sin(\pi - \theta) = \sin\theta$, we replace $\sin\gamma$ with $\sin(\alpha+\beta)$, yielding the compact formula:

$$\boxed{\text{Area} = \frac12 a^{2} \frac{\sin\beta \,\sin(\alpha+\beta)}{\sin\alpha}}.$$

This AAS triangle area equation is what the calculator uses to produce instant results.

Worked Example

Consider a triangle with side $a = 16\ \text{cm}$, $\alpha = 40^\circ$, $\beta = 25^\circ$.

1. Third angle: $\gamma = 180^\circ - 40^\circ - 25^\circ = 115^\circ$.
2. Law of sines constant: $k = \dfrac{16}{\sin 40^\circ} \approx 24.89$.
3. Side b: $b = k \sin 25^\circ \approx 10.53\ \text{cm}$.
4. Side c: $c = k \sin 115^\circ \approx 22.55\ \text{cm}$.
5. Height: $h = a \sin\gamma = 16 \sin 115^\circ \approx 14.50\ \text{cm}$.
6. Area: $\text{Area} = \dfrac12 \times b \times h \approx 76.27\ \text{cm}^2$ (or directly from the formula above).

These outputs match exactly what the solve AAS triangle calculator displays.

How to Use the AAS Triangle Calculator

The interface is designed for ease of use:

• Input fields: Enter side $a$ and the two angles $\alpha$ and $\beta$ (choose either degrees or radians, but stay consistent).
• Output results: The tool immediately returns $\gamma$, $b$, $c$, the height $h$, and the area.
• Precision controls: You can adjust the number of decimal places shown.
• Unit conversion: The calculator works with any length unit—just ensure you interpret the output in the same unit.

Because the tool incorporates the law of sines and angle‑sum logic, you don’t need to perform any algebra yourself. It functions as a dedicated angle angle side calculator and a triangle congruence calculator in one.

Important Considerations

• Angle sum: The sum $\alpha + \beta$ must be less than $180^\circ$ (or $\pi$ rad); otherwise no triangle can exist.
• Angle correspondence: The known side $a$ must be opposite the first supplied angle $\alpha$. If the known side lies between the two angles, the problem is ASA, not AAS.
• Unit consistency: Degrees and radians cannot be mixed. Convert all angles to the same unit before input.
• Obtuse angles: The law of sines works fine with obtuse angles (e.g., $\gamma > 90^\circ$) because $\sin\theta$ remains positive for $\theta$ up to $180^\circ$.

These guidelines help you get accurate results every time.