AAS Triangle Calculator
Solve a triangle given two angles and a non-included side (AAS): find remaining sides, angles & area.
Triangles are one of the most important shapes in mathematics, engineering, and real‑world measurements. From classroom geometry problems to construction layouts and land surveys, triangles help us measure distances, angles, and areas accurately. However, solving a triangle becomes difficult when not all sides and angles are known.
The AAS Triangle Calculator makes this process simple. AAS stands for Angle–Angle–Side, which means you already know two angles and one side of a triangle, where the side is not between the two angles. Using this information, the calculator can instantly find the third angle, the remaining sides, and the area of the triangle.
This tool is designed for students, teachers, engineers, and anyone who needs quick and accurate triangle solutions without doing long manual calculations.
What the AAS Triangle Calculator Is
A Specialized Triangle Solver
The AAS Triangle Calculator is an online trigonometry tool that works specifically with the Angle–Angle–Side triangle case. In this setup, the known side is opposite one of the known angles and is not between the two angles.
Because of this configuration, an AAS triangle always has one unique solution. This makes the calculator reliable for accurate geometry results.
What the Calculator Can Calculate
When you enter the required values, the calculator can determine:
The third angle of the triangle
The lengths of the two unknown sides
The total area of the triangle
Sometimes the perimeter and height
All results are based on standard trigonometry rules used in schools and professional fields.
How the AAS Triangle Calculator Works
Step 1: Enter the Known Values
You start by entering:
Angle A (in degrees)
Angle B (in degrees)
Side a (the side opposite Angle A)
The calculator first checks whether the angles form a valid triangle.
Step 2: Find the Third Angle
The sum of the interior angles of any triangle is always 180 degrees. The calculator uses the angle sum rule:
<br />C = 180^\circ - A - B<br />This gives the value of the third angle.
Step 3: Use the Law of Sines
To calculate the missing sides, the calculator applies the Law of Sines:
From this relationship, the calculator solves for the unknown sides:
<br />b = a \times \frac{\sin B}{\sin A}<br /> <br />c = a \times \frac{\sin C}{\sin A}<br />These formulas link each side to its opposite angle.
Step 4: Calculate the Area
Once two sides and the included angle are known, the area of the triangle can be calculated using:
<br />\text{Area} = \frac{1}{2} \times a \times b \times \sin C<br />This gives the surface area of the triangle in square units.
Key Formulas Used
Angle Sum Formula
<br />A + B + C = 180^\circ<br />This formula ensures that the triangle is valid and helps find the missing angle.
Law of Sines
<br />\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}<br />This rule connects the sides and angles of the triangle.
Side Length Formulas
<br />b = a \times \frac{\sin B}{\sin A}<br /> <br />c = a \times \frac{\sin C}{\sin A}<br />These equations calculate the unknown sides.
Area Formula
<br />\text{Area} = \frac{1}{2} \times a \times b \times \sin C<br />This formula finds the total area of the triangle.
Step‑by‑Step Example
Given Information
Angle A = 40°
Angle B = 60°
Side a = 10 units
Step 1: Find Angle C
<br />C = 180^\circ - 40^\circ - 60^\circ = 80^\circ<br />Step 2: Find Side b
<br />b = 10 \times \frac{\sin 60^\circ}{\sin 40^\circ}<br /> <br />b \approx 13.47 \text{ units}<br />Step 3: Find Side c
<br />c = 10 \times \frac{\sin 80^\circ}{\sin 40^\circ}<br /> <br />c \approx 15.30 \text{ units}<br />Step 4: Find the Area
<br />\text{Area} = \frac{1}{2} \times 10 \times 13.47 \times \sin 80^\circ<br /> <br />\text{Area} \approx 66.3 \text{ square units}<br />The calculator completes all these steps instantly.
Features of the AAS Triangle Calculator
Fast and Accurate Results
The calculator produces instant answers using verified trigonometric formulas. This eliminates human calculation errors and saves time for users.
Simple and User‑Friendly Design
The input fields are clearly labeled. Even beginners can use the tool without confusion.
Automatic Error Checking
The calculator checks angle values to make sure they form a valid triangle before showing results.
Complete Triangle Data
Users receive all important triangle measurements at once, including angles, sides, and area.
Uses and Applications
Education and Homework Support
Students often struggle with trigonometry and triangle problems. This calculator helps them verify answers and understand how angles and sides are related. Teachers can also use it to demonstrate triangle solving in class.
Engineering and Construction
Engineers rely on triangle calculations for structural designs, roof angles, and support systems. When only partial measurements are available, the AAS calculator helps find the missing values accurately.
Surveying and Land Measurement
Surveyors use angles and distances to measure land. AAS calculations help them determine unknown sides and areas, making mapping and boundary planning more precise.
Helpful Tips
Use Degrees for Angles
Always enter angles in degrees, not radians.
Check Angle Total
The two known angles must add up to less than 180°.
Match Side with Angle
Side a must be opposite Angle A for correct results.
Keep Units Consistent
If the side is in meters, all outputs will also be in meters.
Common Mistakes
Confusing AAS with ASA
In AAS, the known side is not between the two angles.
Entering Invalid Angles
Angles like 95° and 90° cannot form a triangle.
Using the Wrong Side
If the side does not match the angle, the results will be incorrect.
FAQs
What does AAS stand for?
AAS means Angle–Angle–Side.
Is an AAS triangle always solvable?
Yes, it always has one unique solution.
Can the calculator find the area?
Yes, it uses trigonometric area formulas.
Is this tool free to use?
Most AAS triangle calculators are available for free online.
Final Words
The AAS Triangle Calculator is a powerful and easy‑to‑use tool for solving triangles. It saves time, improves accuracy, and removes confusion from geometry problems.
Whether you are a student, teacher, engineer, or surveyor, this calculator helps you solve triangles quickly using just two angles and one side.