ASA Triangle Calculator

Solve a triangle given two angles and the included side (Angle-Side-Angle) using the law of sines.

Enter Two Angles & Included Side

ASA: Given angle A, side c (between A and B), and angle B. The calculator finds angle C, sides a, b, area, and perimeter.

Complete Triangle

Area
38.30222
square units
Angle A 50 deg
Angle B 60 deg
Angle C 70 deg
Side a (opposite A) 8.152156
Side b (opposite B) 9.211398
Side c (included) 10
Perimeter 27.363554
Area 38.30222

Step-by-Step Solution

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

What Is an ASA Triangle?

An ASA (Angle-Side-Angle) triangle is a triangle where two angles and the side between them (the included side) are known. This gives enough information to uniquely determine the triangle -- meaning there is exactly one triangle that satisfies the given conditions. This is one of the triangle congruence criteria taught in geometry.

The ASA configuration is solved by first finding the third angle (since all angles in a triangle sum to 180 degrees), and then using the law of sines to find the remaining sides.

How to Solve an ASA Triangle

Step 1: Find the Third Angle

Since the sum of all angles in a triangle is 180 degrees: C = 180 - A - B. If C is not positive, the given angles do not form a valid triangle.

Step 2: Use the Law of Sines

The law of sines states: a/sin(A) = b/sin(B) = c/sin(C). Since we know c, A, and B (and now C), we can find sides a and b:

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

Step 3: Calculate the Area

Using the SAS formula with any two known sides and their included angle, or using: Area = 1/2 x a x b x sin(C).

Triangle Congruence Cases

ASA (Angle-Side-Angle)

Two angles and the included side. Always gives a unique triangle.

C = 180 - A - B

AAS (Angle-Angle-Side)

Two angles and a non-included side. Also gives a unique triangle.

Equivalent to ASA after finding the third angle.

SAS (Side-Angle-Side)

Two sides and the included angle.

Use Law of Cosines for third side.

SSS (Side-Side-Side)

All three sides known.

Use Law of Cosines for angles.

SSA (Side-Side-Angle)

Two sides and a non-included angle. May give 0, 1, or 2 solutions.

Ambiguous case of Law of Sines.

Law of Sines

Fundamental relationship in any triangle.

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

The Law of Sines

The Law of Sines establishes a proportional relationship between each side of a triangle and the sine of its opposite angle. It states that in any triangle, the ratio of a side to the sine of its opposite angle is constant:

a / sin(A) = b / sin(B) = c / sin(C) = 2R

where R is the circumradius (radius of the circumscribed circle). This law is the primary tool for solving ASA and AAS triangles.

Worked Example

Solve a triangle with A = 40 degrees, c = 12, B = 75 degrees:

  1. C = 180 - 40 - 75 = 65 degrees
  2. a = 12 x sin(40) / sin(65) = 12 x 0.6428 / 0.9063 = 8.510
  3. b = 12 x sin(75) / sin(65) = 12 x 0.9659 / 0.9063 = 12.789
  4. Area = 1/2 x 8.510 x 12.789 x sin(65) = 49.32 square units

Common Mistakes to Avoid

  • Forgetting to check that A + B is less than 180 degrees (otherwise no valid triangle exists).
  • Confusing the included side -- in ASA, the known side must be between the two known angles.
  • Using the wrong angle in the law of sines -- each side is opposite its corresponding angle.
  • Mixing up degrees and radians in calculations.

Practical Applications

  • Surveying: Triangulation uses ASA to determine distances when angles can be measured from two known points.
  • Navigation: Determining position using angle measurements from two landmarks at a known distance apart.
  • Astronomy: Calculating stellar distances using parallax measurements (essentially ASA triangulation).
  • Architecture: Designing roof trusses and structural supports with known angle constraints.