AAS Triangle Calculator

Enter two angles and the side opposite one of them (the AAS case) to solve the triangle — the missing angle, both remaining sides, and the area — using the Law of Sines.

Quick Facts

Method
Law of Sines: a/sin A = b/sin B = c/sin C
The third angle is always 180° − A − B, since a triangle's angles sum to 180°.

Your Results

Calculated
Angle C
-
180° − A − B
Side b
-
Opposite angle B
Side c
-
Opposite angle C
Area
-
½ · a · b · sin C, square units

Ready

Enter angles A and B and the length of side a, then press Calculate.

Formula and method for the AAS triangle case

AAS ("Angle-Angle-Side") is one of the standard triangle-solving cases in trigonometry: you know two angles and the length of a side that is not between them — it sits opposite one of the two known angles. That is enough information to determine every remaining angle and side, because a triangle's three angles always sum to 180° and the Law of Sines links every side to the sine of its opposite angle.

How the calculation works

With angle A, angle B, and side a (opposite angle A) known, the solution proceeds in two steps. First, the missing angle: C = 180° − A − B. Second, the Law of Sines states that the ratio of a side to the sine of its opposite angle is the same for all three sides of a triangle:

a / sin(A) = b / sin(B) = c / sin(C)

Rearranging that relationship for the two unknown sides gives b = a × sin(B) / sin(A) and c = a × sin(C) / sin(A). Once all three sides and angles are known, the triangle's area follows from Area = ½ × a × b × sin(C) (equivalent to the familiar ½ × base × height, using two known sides and the angle between them).

Common mistakes

  • Angle sum too large: the two given angles must add to less than 180°, or no triangle exists — there would be nothing left for the third angle.
  • Confusing AAS with ASA: in ASA the given side is between the two known angles; in AAS it is opposite one of them. Both cases solve the same way once you compute the third angle, but mixing up which side you were given leads to labeling the wrong side as "a."
  • Degrees vs. radians: this calculator expects angles in degrees. Entering a radian value (like 0.87 instead of 50) will produce a nonsensical triangle.
  • Mismatched units: the output sides are in whatever unit you entered for side a (inches, cm, meters, and so on) — mixing units elsewhere in a larger project will throw off downstream area or material calculations.

Real-world applications

  • Surveying and navigation use AAS to find distances across land or water when two bearing angles and one known baseline are measurable but the far side is not directly reachable
  • Engineering and construction use it to check truss and roof-truss geometry when angle specifications and one member length are given
  • Astronomy and geodesy use angle-based triangulation (AAS/ASA logic) to compute distances that cannot be measured directly
  • Trigonometry coursework uses AAS as one of the four standard triangle-solving cases, alongside SAS, SSS, and the ambiguous SSA

Frequently Asked Questions

What does AAS mean in triangle solving?
AAS stands for Angle-Angle-Side: you know two angles of a triangle and the length of a side that is not between them (it is opposite one of the two known angles). This is enough information to solve the whole triangle, because knowing two angles automatically gives you the third (angles sum to 180 degrees), and the Law of Sines then finds the remaining sides.
What formula solves an AAS triangle?
First find the missing angle with C = 180° − A − B. Then apply the Law of Sines, a/sin(A) = b/sin(B) = c/sin(C), to solve for the two unknown sides: b = a × sin(B)/sin(A) and c = a × sin(C)/sin(A). The triangle's area is ½ × a × b × sin(C).
Why must the two given angles sum to less than 180 degrees?
Every triangle's interior angles must add up to exactly 180 degrees. If your two given angles already total 180 or more, there is no room left for a third angle, so no such triangle can exist. The calculator checks this and flags an invalid input if angle A plus angle B is not less than 180 degrees.
Does AAS give one unique triangle?
Yes. Unlike the ambiguous SSA case, AAS always determines a unique triangle. Once both angles and one side are fixed, every other angle and side follows directly from the Law of Sines, with no second possible solution.

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