Aas Triangle Calculator

Background.

Know two angles of a triangle and you already know the third, because the three always add up to 180°. Two angles therefore fix a triangle's shape completely. What they leave open is its size, and a single side length settles that. This is why angle-angle-side, or AAS, is one of the congruence rules of school geometry: two triangles that share two angles and a correspondingly placed side are identical.

Turning that guarantee into numbers is the work of the law of sines, which says that in any triangle each side divided by the sine of the angle opposite it gives the same value. Know one side and the angle facing it and you know that shared value; every other side is then that value multiplied by the sine of its own opposite angle. Here you enter side a, the angle A across from it and a second angle B, and the page returns side b, which faces B, together with the third angle C.

The pairing is the part people get wrong. Side a must be the side facing angle A, not one of the two sides that meet at A. If what you actually have is two angles with the known side between them, the angle-side-angle case typical of surveying, where a baseline is measured and the sight line to a distant point is read from each end, subtract the two angles from 180° first. The result is the angle facing the baseline, and that is your A.

One side comes out per run; for the remaining side, run it again with C as the target angle. Two sides plus an angle that is not between them (SSA) is a different problem, because those data can fit two different triangles or none, and this page does not attempt it.

What is aas triangle calculator?

An AAS triangle problem supplies two angles and a side that does not lie between them, and asks for the remaining parts. Because the angles of a triangle sum to 180°, the third angle follows by subtraction, and because a/sin A = b/sin B = c/sin C holds in every triangle, each unknown side is the known side scaled by a ratio of sines. The data always fix exactly one triangle, provided the two given angles add up to less than 180°. Lengths come out in whatever unit side a was measured in.

How to use this calculator.

1. 01Label the triangle so the side you know is a and the angle across the triangle from it is A. Enter the length in Known side a, in any unit.
2. 02Enter that opposite angle in Opposite angle A, in degrees. If your angle is in radians, multiply by 180/π first; one radian is about 57.2958°.
3. 03If your known side lies between the two angles you measured, subtract those angles from 180° and use the result as angle A.
4. 04Enter the angle facing the side you want in Target angle B. A and B together must be less than 180°, or no triangle exists and the page will say so.
5. 05Read Solved side b in the same unit as side a, and the Third angle in degrees. For the last side, run the page again with the third angle as the target.

The formula.

Two relations do all the work. The angle sum gives C = 180° − A − B. The law of sines, a/sin A = b/sin B, rearranges to b = a × sin B ÷ sin A. The shared ratio a/sin A is the diameter of the circle that passes through all three corners, so the law amounts to saying that every side equals that diameter times the sine of the angle facing it. Angles are converted from degrees to radians before the sines are taken. The answer is most sensitive to angle A, because sin A sits in the denominator: when A is small its sine is small, and with a = 5 and B = 60°, entering A as 5° instead of 6° makes b about 20% larger. When A is large the result barely moves for the same one-degree slip. Angle pairs that reach 180° are refused, since they would leave a third angle of zero or less.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.