Solve a triangle from two angles and the non-included side (AAS: Angle-Angle-Side). Enter angles A and B and side a (the side opposite angle A). The calculator finds the third angle, all remaining sides, the perimeter, and the area using the Law of Sines.
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Standard formula
Law of Sines: a / sin(A) = b / sin(B) = c / sin(C). Angle sum: A + B + C = 180 degrees.
1. Known Angles
deg
deg
2. Known Side
units
Triangle Solution
Angle C
-
Third angle (degrees)
Side b
-
Opposite angle B
Side c
-
Opposite angle C
Area
-
Square units
All Triangle Measurements
Angle A-
Angle B-
Angle C-
Side a (opposite A)-
Side b (opposite B)-
Side c (opposite C)-
Perimeter-
Additional Properties
Area-
Law of Sines ratio (a/sin A)-
Triangle type by angles-
Triangle type by sides-
Circumradius R-
Inradius r-
Height from side a-
Worked Example (default inputs)
Given: Angle A = 50 deg, Angle B = 60 deg, side a = 10 units
Step 1 - Third angle: C = 180 - 50 - 60 = 70 deg
Step 2 - Law of Sines ratio: a / sin(A) = 10 / sin(50 deg) = 10 / 0.76604 = 13.0541
Step 3 - Side b: b = 13.0541 x sin(60 deg) = 13.0541 x 0.86603 = 11.3052 units
Step 4 - Side c: c = 13.0541 x sin(70 deg) = 13.0541 x 0.93969 = 12.2668 units
Step 5 - Area: Area = (1/2) x a x b x sin(C) = 0.5 x 10 x 11.3052 x sin(70 deg) = 0.5 x 10 x 11.3052 x 0.93969 = 53.1169 sq units
AAS (Angle-Angle-Side) is a triangle congruence and solution method that uses two interior angles and a side that is not between those angles. Because the angles of any triangle must sum to exactly 180 degrees, knowing two angles immediately fixes the third. Once all three angles are known, the Law of Sines provides the remaining sides. AAS always has exactly one solution, which makes it simpler than the SSA case, which can produce zero, one, or two solutions.
The Formula
Given angle A, angle B, and side a (opposite angle A):
Step
Formula
Description
1
C = 180 - A - B
Find the third angle
2
b = a x sin(B) / sin(A)
Find side b via Law of Sines
3
c = a x sin(C) / sin(A)
Find side c via Law of Sines
4
Perimeter = a + b + c
Sum of all sides
5
Area = (a x b x sin(C)) / 2
Area using two sides and included angle
AAS vs ASA
ASA (Angle-Side-Angle) places the known side between the two known angles, whereas AAS places the known side opposite one of the known angles. Both are valid congruence conditions and both produce a unique triangle. The distinction matters when labelling diagrams, but the solving method (Law of Sines) is the same for both once the third angle is found.
The Law of Sines
The Law of Sines states that for any triangle with angles A, B, C and opposite sides a, b, c: a / sin(A) = b / sin(B) = c / sin(C). This common ratio equals twice the circumradius (2R) of the triangle. The law applies to all triangles, not just right-angled ones, making it the standard tool for solving AAS and ASA triangles.
When AAS Has No Solution
AAS fails to produce a valid triangle if the two given angles sum to 180 degrees or more (leaving zero or a negative value for the third angle), or if a given angle is zero or negative. The given side must also be a positive number. This calculator displays an error message in those cases rather than returning nonsensical results.
Sources and method: Law of Sines derivation from standard trigonometry references. Angle sum property: the interior angles of any Euclidean triangle sum to 180 degrees. AAS congruence theorem: two triangles are congruent if two angles and a non-included side of one are equal to the corresponding parts of the other.
This calculator works in Euclidean (flat) geometry. Results are rounded to four decimal places for display. For surveying, construction, or navigation applications, verify results with a licensed professional.
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