ASA Triangle Calculator (Angle-Side-Angle)
Enter two angles and the side between them (the included side). The calculator finds the third angle using the angle-sum rule, then applies the Law of Sines to compute all remaining sides. It also gives you the area, perimeter, semi-perimeter, inradius, and circumradius, plus a live triangle diagram and step-by-step working.
Formula
Worked example
Given A = 60°, c = 10, B = 70°: C = 180 - 60 - 70 = 50°. Then a = 10 * sin(60°) / sin(50°) = 10 * 0.8660 / 0.7660 = 11.305. And b = 10 * sin(70°) / sin(50°) = 10 * 0.9397 / 0.7660 = 12.268. Area = 0.5 * 11.305 * 12.268 * sin(50°) = 53.21. Perimeter = 11.305 + 12.268 + 10 = 33.573. Inradius r = 53.21 / (33.573 / 2) = 3.170. Circumradius R = (11.305 * 12.268 * 10) / (4 * 53.21) = 6.529.
What is an ASA triangle?
ASA stands for Angle-Side-Angle. It describes a triangle where you know two angles and the one side that lies directly between them (called the included side). Because a triangle has three angles that always add up to 180 degrees, knowing two of them immediately gives you the third. Once you have all three angles and one side, the Law of Sines lets you calculate the two remaining sides. This means an ASA triangle is always uniquely determined: there is exactly one triangle shape that satisfies any valid set of ASA values.
How the ASA formulas work
First, the missing angle C is found with the angle-sum rule: C = 180 deg - A - B (or pi - A - B in radians). Next, the Law of Sines states that the ratio of any side to the sine of its opposite angle is constant across the whole triangle. Given the included side c, you get a = c * sin(A) / sin(C) and b = c * sin(B) / sin(C). With all three sides known, the area follows from the cross-product formula Area = (1/2) * a * b * sin(C). The semi-perimeter s = (a + b + c) / 2 then gives the inradius r = Area / s and the circumradius R = (a * b * c) / (4 * Area). Every output in this calculator is derived from those six formulas in that exact order.
When ASA applies - and when it does not
ASA applies whenever you measure two angles at the ends of a known side. A common example in surveying is triangulation: you set two stations a known distance apart and measure the bearing to a distant target from each station. The two bearing angles and the known baseline are your A, B and c. The ASA configuration always produces a unique triangle, unlike SSA (two sides and a non-included angle), which can yield zero, one, or two solutions. If the two given angles already sum to 180 degrees or more, no valid triangle exists and this calculator returns no result.
Inradius, circumradius and triangle circles
Every triangle has two special circles. The incircle is the largest circle that fits entirely inside the triangle; its radius (the inradius, r) equals the area divided by the semi-perimeter. The circumcircle passes through all three vertices; its radius (the circumradius, R) equals the product of all three sides divided by four times the area. For an equilateral triangle, R is always exactly twice r. For any triangle, r / R equals 4 * sin(A/2) * sin(B/2) * sin(C/2), which is a useful cross-check. The ratio also tells you how "round" the triangle is: a circle has r = R, and more elongated or thin triangles have r much smaller than R.
Triangle type classification
| Classification | Criterion | Notes |
|---|---|---|
| Acute | All angles < 90 deg | All altitudes fall inside the triangle |
| Right | One angle = 90 deg | Pythagorean theorem applies: a^2 + b^2 = c^2 |
| Obtuse | One angle > 90 deg | One altitude falls outside the triangle |
| Equilateral | All sides equal | All angles are 60 deg |
| Isosceles | Two sides equal | Base angles are equal |
| Scalene | All sides different | All angles are different |
How a triangle is classified by its angles and sides.
Frequently asked questions
What does ASA mean in geometry?
ASA stands for Angle-Side-Angle. It describes the case where you know two angles of a triangle and the one side that is sandwiched directly between them. That configuration uniquely determines the triangle, meaning there is exactly one triangle (up to reflection and rotation) that satisfies any valid ASA input.
How do you find the missing side in an ASA triangle?
First compute the third angle: C = 180 deg - A - B. Then apply the Law of Sines. If c is the included side, side a (opposite A) equals c * sin(A) / sin(C), and side b (opposite B) equals c * sin(B) / sin(C). This calculator does both steps automatically.
What is the Law of Sines and why does it work for ASA?
The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C) for any triangle. It works because all three expressions equal the diameter of the circumscribed circle (2R). In the ASA case you know c, A and B (from which you derive C), so you can isolate a and b by cross-multiplying.
Can two different triangles share the same ASA values?
No. Unlike the SSA case, ASA is a congruence criterion: any two triangles with the same pair of angles and the same included side are congruent. That is why ASA always produces exactly one solution as long as the angles are positive and sum to less than 180 degrees.
What happens if the two angles already add up to 180 degrees or more?
If A + B is equal to or greater than 180 degrees, the third angle would be zero or negative, which is impossible in a real triangle. The calculator returns no result in that case. Make sure each angle is a positive number and that their sum leaves room for a positive third angle.
How is the area of an ASA triangle calculated?
Once you have all three sides (via the Law of Sines), the area is computed as (1/2) * a * b * sin(C). An equivalent formula that works directly from the ASA inputs is Area = (c^2 / 2) * sin(A) * sin(B) / sin(A + B), which avoids computing the sides first.
What is the difference between inradius and circumradius?
The inradius (r) is the radius of the incircle, the largest circle that fits inside the triangle. It equals the area divided by the semi-perimeter. The circumradius (R) is the radius of the circumcircle, the circle that passes through all three vertices. It equals the product of all three sides divided by four times the area. For an acute triangle the circumcenter is inside the triangle; for an obtuse triangle it is outside.