AAS Triangle Calculator (Angle-Angle-Side)
Enter two angles and the non-included side of a triangle (AAS) to solve for every remaining measurement. The calculator applies the angle-sum rule to find the third angle, then uses the Law of Sines to compute the missing sides. Results include area, perimeter, semi-perimeter, the inscribed circle radius, and the circumscribed circle radius, with a live triangle diagram and full step-by-step working.
Formula
Worked example
Given A = 40 deg, B = 25 deg, a = 16 cm: C = 180 - 40 - 25 = 115 deg. Law of Sines ratio: 16 / sin(40) = 24.888. Side b = 24.888 x sin(25) = 10.513 cm. Side c = 24.888 x sin(115) = 22.548 cm. Area by Heron = 76.27 cm^2. Inradius r = 76.27 / 24.530 = 3.108 cm. Circumradius R = 16 / (2 x sin 40) = 12.444 cm.
What is the AAS congruence condition?
AAS stands for Angle-Angle-Side. A triangle is uniquely determined when you know two of its interior angles and one side that is NOT between those two angles. Because the three interior angles of any triangle must add up to 180 degrees, knowing two angles immediately gives you the third. Once all three angles are known, the Law of Sines ties every side to the same ratio, so one known side length fixes the scale of the whole triangle. The AAS case is different from the ASA (Angle-Side-Angle) case only in which side you are given: in ASA the known side is between the two known angles; in AAS it is opposite one of them. Both cases are fully solvable and produce exactly one triangle (assuming the angles sum to less than 180 degrees and all values are positive).
How to solve an AAS triangle step by step
Step 1 - Find the third angle: subtract the two known angles from 180 degrees, so C = 180 - A - B. If the result is zero or negative, the inputs are invalid. Step 2 - Apply the Law of Sines to find the missing sides. The law states that a / sin(A) = b / sin(B) = c / sin(C), where a is the side opposite angle A, and so on. Rearranging gives b = a x sin(B) / sin(A) and c = a x sin(C) / sin(A). Step 3 - Compute the area using Heron's formula: find the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). Step 4 - Compute the inradius r = Area / s and the circumradius R = a / (2 x sin(A)).
Inradius and circumradius: what they mean
Every triangle has two special associated circles. The inscribed circle (incircle) is the largest circle that fits entirely inside the triangle, touching all three sides. Its radius, the inradius r, equals the area of the triangle divided by its semi-perimeter: r = K / s. The circumscribed circle (circumcircle) passes through all three vertices. Its radius, the circumradius R, can be found directly from the Law of Sines: R = a / (2 x sin A) = b / (2 x sin B) = c / (2 x sin C). A large circumradius relative to the inradius indicates a very elongated or thin triangle; for an equilateral triangle the ratio R/r is exactly 2.
AAS vs. ASA vs. SSA: choosing the right solver
Choose your solver based on which three measurements you actually know. AAS (this calculator) - two angles and the side opposite one of them. ASA - two angles and the side between them; the math is the same once you find the third angle. SAS - two sides and the included angle; use the Law of Cosines. SSS - all three sides; use the Law of Cosines or Heron's formula. SSA (ambiguous case) - two sides and a non-included angle; there can be zero, one, or two valid triangles, which requires extra care. The AAS and ASA cases are the most straightforward because the Law of Sines applies cleanly with no ambiguity.
Triangle type classification by angles
| Largest angle | Triangle type | Property |
|---|---|---|
| All angles < 90 deg | Acute | All altitudes lie inside the triangle |
| Exactly one angle = 90 deg | Right | Pythagorean theorem applies; hypotenuse is longest side |
| One angle > 90 deg | Obtuse | Altitude from the obtuse vertex falls outside the triangle |
| All angles = 60 deg | Equilateral (special case) | Also equilateral; all sides and angles equal |
Knowing the largest angle of a triangle determines whether it is acute, right, or obtuse.
Frequently asked questions
What is the difference between AAS and ASA triangles?
Both AAS and ASA give you two angles and one side. The difference is the position of the known side relative to the known angles. In ASA the known side is between (included by) the two known angles. In AAS the known side is opposite one of the angles, not between them. Mathematically both cases lead to the same solution method once you use the angle-sum rule to find the third angle, so the numerical procedure is identical.
Can an AAS triangle have no solution?
Yes, in two situations: first, if the two given angles sum to 180 degrees or more, no triangle exists because the third angle would be zero or negative. Second, if any entered value is zero or negative (a negative side length or a zero angle is geometrically impossible). When the inputs are valid (angles sum to less than 180 degrees, side is positive) there is exactly one triangle that satisfies the AAS conditions, with no ambiguity.
How is the area of an AAS triangle calculated?
Once you have all three sides (a, b, c) from the Law of Sines, you can use Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s x (s-a) x (s-b) x (s-c)). Alternatively, once you know all three angles and one side, you can use the formula Area = (a^2 / 2) x sin(B) x sin(A + B) / sin(A), which is equivalent and avoids computing the other sides first.
What is the Law of Sines and why does it work for AAS?
The Law of Sines states that in any triangle, the ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This constant equals the diameter of the circumscribed circle (2R). The law works for AAS because knowing all three angles (the third found from 180 - A - B) and one side fixes that common ratio, which then gives every other side directly by rearranging the equation.
How do I verify my AAS triangle solution?
Three quick checks confirm the solution. First, verify that all three angles sum to 180 degrees. Second, verify that the Law of Sines ratio is constant: a/sin(A) should equal b/sin(B) and c/sin(C) to the precision of your inputs. Third, use the formula Area = 0.5 x b x c x sin(A) and compare it with Heron's formula output - both should match. If any check fails, re-examine whether the correct side (the non-included one) was entered opposite the correct angle.