AAS (Angle-Angle-Side) Triangle Calculator

Formula

To calculate the length of the missing side (a):

\[ a = \frac{b \cdot \sin(A)}{\sin(B)} \]

Where:

\(a\) is the length of the missing side
\(b\) is the length of a known side
\(A\) is the measure of the known angle (in radians)
\(B\) is the measure of the other known angle (in radians)

What is AAS (Angle-Angle-Side)?

AAS (Angle-Angle-Side) is a rule used to prove that two triangles are congruent. According to this rule, if two angles and the non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the two triangles are congruent. The term “non-included side” refers to the side that is not between the two angles. This rule is a shortcut derived from the ASA (Angle-Side-Angle) postulate, where the third sides and angles of the two triangles are proven to be congruent by the laws of geometry. The AAS rule is particularly useful in geometry problems where it is not immediately obvious that two triangles are congruent.

Example Calculation

Let's assume the following values:

Length of known side (\(b\)) = 5 units
Measure of known angle (\(A\)) = 0.785 radians (45 degrees)
Measure of other known angle (\(B\)) = 0.524 radians (30 degrees)

Using the formula:

\[ a = \frac{5 \cdot \sin(0.785)}{\sin(0.524)} = \frac{5 \cdot 0.7071}{0.5} = \frac{3.5355}{0.5} = 7.071 \text{ units} \]

The length of the missing side is approximately 7.071 units.