ASA Triangle Calculator

Enter two angles and the length of the side between them (ASA) to solve the triangle: the third 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° minus the two given angles; ASA fixes a triangle uniquely, unlike the ambiguous SSA case.

Your Results

Calculated
Angle C
-
Third angle (180° − A − B)
Side a
-
Opposite Angle A
Side b
-
Opposite Angle B
Area
-
½ × a × b × sin(C)

Ready

Enter two angles and the included side, then press Calculate.

How the ASA Triangle Calculator works

ASA (Angle-Side-Angle) is one of the standard ways to fully define a triangle: two interior angles, A and B, plus the length of the side that connects them (the included side, c). Because a triangle's three angles always sum to 180 degrees and the Law of Sines ties every side to the sine of its opposite angle, knowing A, B, and c is enough to solve for everything else, with exactly one valid triangle as the result.

The formulas

  • Third angle: C = 180° − A − B
  • Law of Sines: a / sin(A) = b / sin(B) = c / sin(C)
  • Side a (opposite angle A): a = c × sin(A) / sin(C)
  • Side b (opposite angle B): b = c × sin(B) / sin(C)
  • Area: Area = ½ × a × b × sin(C), equivalent to c² × sin(A) × sin(B) / (2 × sin(C))
  • Perimeter: a + b + c

Enter the two known angles in degrees and the length of the included side in any consistent unit (inches, cm, meters — the calculator does not assume one). It converts the angles to radians internally for the trigonometry, then reports the third angle, the two unknown sides in the same unit as c, and the area in that unit squared.

Why ASA always has exactly one solution

Unlike SSA (side-side-angle), which can produce zero, one, or two valid triangles depending on the numbers (the "ambiguous case"), ASA is never ambiguous. Fixing two angles fixes the triangle's shape completely, and the included side fixes its scale — there is exactly one triangle that satisfies the inputs, provided A and B are each positive and sum to less than 180°.

Common mistakes

  • Angles in the wrong unit: the calculator expects degrees, not radians. Entering 1.05 instead of 60 produces a near-zero, meaningless triangle.
  • Wrong included side: ASA specifically needs the side between the two named angles. If you actually know the side opposite one of the angles instead, that is AAS, not ASA — the Law of Sines formulas above still apply, but the solving order differs slightly.
  • Angles that leave no room for a third: if A + B is 180° or more, no triangle exists; double-check your angle measurements.

Real-world applications

  • Surveying and land measurement, where two bearing angles and a baseline distance locate a remote point (triangulation)
  • Navigation and astronomy, sighting two angles from a known baseline to fix a third position
  • Engineering and construction, laying out trusses, roof pitches, or brackets from angle-and-length specifications
  • Trigonometry coursework, since ASA is one of the standard triangle congruence and solving cases alongside SSS, SAS, and AAS

Frequently Asked Questions

What does ASA mean in triangle solving?
ASA stands for Angle-Side-Angle: you know two interior angles of a triangle and the length of the side between them, called the included side. Because two angles fix the triangle's shape and the included side fixes its scale, ASA data determines exactly one triangle, with no ambiguity.
How is the third angle found?
The interior angles of any triangle sum to 180 degrees, so the third angle equals 180° minus the two known angles: C = 180° − A − B. This must come out positive, which means A and B together must sum to less than 180°.
What formula finds the two unknown sides?
The Law of Sines: a / sin(A) = b / sin(B) = c / sin(C). Once the third angle C is known, the remaining sides are a = c × sin(A) / sin(C) and b = c × sin(B) / sin(C), where c is the included side you entered.
Can any two angles and a side always form a valid triangle?
No. Each angle must be greater than 0 degrees, and the two given angles must sum to less than 180 degrees so a positive third angle remains. If A + B is 180 degrees or more, those inputs cannot form a triangle.

Related calculators