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AAS Triangle Solver

Enter angles A and B and side a (opposite A) to solve for angle C, sides b and c, and the area.

Result

80.00°
13.47
15.32
66.34

How it is calculated

You enter angles A and B in degrees and side a, the side opposite angle A. The third angle follows from the angle sum: C = 180° − A − B. The law of sines then gives the other sides: b = a × sin B ÷ sin A and c = a × sin C ÷ sin A. The area uses two sides and the angle between them: area = ½ × a × b × sin C.

Angles must be above zero and A + B must be less than 180°, otherwise there is no triangle and the tool returns nothing; side a must be positive. One angle may be obtuse, as long as the sum stays under 180°. Sides come out in whatever length unit you used for a, and the area in the matching square unit. Results are rounded to two decimals, and angle C is shown with a degree sign.

Worked example

Take A = 40°, B = 60° and a = 10. Then C = 180 − 40 − 60 = 80.00°. With sin 40° = 0.6428, sin 60° = 0.8660 and sin 80° = 0.9848: b = 10 × 0.8660 ÷ 0.6428 = 13.47 and c = 10 × 0.9848 ÷ 0.6428 = 15.32. Area = ½ × 10 × 13.47 × 0.9848 = 66.34 square units. Check: the longest side, c = 15.32, is opposite the largest angle, C = 80°, as it should be.

Try this example →

Common mistakes

  • Entering the side that lies opposite angle B (or C) in the side a field. The tool needs the side opposite angle A, so label the triangle so that a faces A.
  • Entering angles in radians. The inputs are degrees; 1.05 would be read as about one degree.
  • Entering two angles that add up to 180° or more, such as 100 and 80. No triangle exists and the tool returns no result.
  • Typing the third angle instead of two angles and a side. The tool already computes C, so give A and B only.
  • Mixing units, for example a in centimetres but reading the area as square metres.

About this calculator

The angle-angle-side (AAS) case is one of the standard triangle-solving scenarios in trigonometry: you know two angles and a side that is not between them (non-included side). Since the two known angles determine the third automatically (angles sum to 180°), the Law of Sines — a/sin A = b/sin B = c/sin C — can then be used directly to find the two remaining sides, and the area follows from the standard triangle-area formula.

This case is guaranteed to produce exactly one unique triangle (unlike the ambiguous SSA case), making it a reliable building block in surveying, navigation, and geometry problems where two angles and one measured length are easier to obtain than three sides. Students and engineers use it whenever a full angle-side-angle layout isn't available but a non-included side is.

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