Triangle Calculator (ASA) — One Side & Two Angles

Last updated: 2026-05-18

Solving a Triangle from One Side and Two Angles

The ASA (Angle-Side-Angle) case specifies one side together with the two angles at its endpoints. Since the three angles of any triangle sum to 180°, the third angle is immediately determined. From there, the Law of Sines yields the remaining sides, and a direct formula from the extended Law of Sines gives the circumradius.

Third angle

$C = 180° - A - B$

This is exact arithmetic — no trigonometry is needed yet. It also means that knowing any two angles determines the triangle's shape completely; the side length sets its scale.

Unknown sides (Law of Sines)

The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

With the known side $c$ and the derived angle $C$:

$a = \frac{c \sin A}{\sin C}, \qquad b = \frac{c \sin B}{\sin C}$

Worked example — $c = 8$, $A = 50°$, $B = 60°$:

$C = 180° - 50° - 60° = 70°$

$a = \frac{8 \sin 50°}{\sin 70°} \approx \frac{8 \times 0.766}{0.940} \approx 6.527$

$b = \frac{8 \sin 60°}{\sin 70°} \approx \frac{8 \times 0.866}{0.940} \approx 7.378$

Area

The area can be computed directly from the ASA inputs without finding the sides first:

$S = \frac{c^2 \sin A \sin B}{2 \sin C}$

This is exact and avoids rounding from the intermediate side computations.

Circumradius from the extended Law of Sines

The common ratio in the Law of Sines equals twice the circumradius:

$\frac{c}{\sin C} = 2R \implies R = \frac{c}{2\sin C}$

For the example above: $R = 8 \div (2\sin 70°) \approx 4.257$.

For a right triangle ($C = 90°$, $\sin 90° = 1$): $R = c/2$ — the circumradius is half the hypotenuse. This is a standard result that follows directly from the extended Law of Sines.

Inradius

$r = \frac{S}{s}, \quad s = \frac{a + b + c}{2}$

ASA vs. AAS

Both configurations uniquely determine a triangle:

• ASA: the known side is between the two angles (the side is an included side).
• AAS: the known side is not between the two angles.

Numerically, both are solved identically: find $C$, then apply the Law of Sines. The distinction exists in Euclidean geometry proofs of congruence but does not affect computation.

The ambiguous (SSA) case

The Law of Sines has an ambiguous case when applied to SSA (two sides and a non-included angle): two different triangles can satisfy the same measurements. The ASA configuration avoids that issue, because fixing the angles at both ends of the known side determines a single triangle.