Law of Sines Calculator — AAS Triangle Solver

Last updated: 2026-05-19

What This Calculator Solves

The Law of Sines relates every side of a triangle to the sine of its opposite angle. Given two angles and the side opposite one of them (the AAS configuration), it determines the third angle, both unknown sides, the area, and the radius of the circumscribed circle.

The Law of Sines

For any triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

The common ratio equals the diameter of the circumscribed circle. This single identity connects all three sides to their angles and explains why the circumradius $R$ appears naturally.

Solving an AAS triangle

Given: angle $A$, side $a$ (opposite $A$), angle $B$.

Step 1: Find the third angle

$C = 180° - A - B$

Two angles fix the shape of the triangle entirely; the known side sets its scale.

Step 2: Find the unknown sides

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

Step 3: Find the area

$S = \frac{1}{2} \cdot a \cdot b \cdot \sin C$

Step 4: Find the circumradius

$R = \frac{a}{2 \sin A}$

Worked Example

Inputs: $A = 30°$, $a = 10$, $B = 45°$

$C = 180° - 30° - 45° = 105°$

$b = \frac{10 \sin 45°}{\sin 30°} = \frac{10 \cdot \tfrac{\sqrt{2}}{2}}{\tfrac{1}{2}} = 10\sqrt{2} \approx 14.142$

$c = \frac{10 \sin 105°}{\sin 30°} = \frac{10 \times 0.9659}{0.5} \approx 19.319$

$S = \tfrac{1}{2} \times 10 \times 14.142 \times \sin 105° \approx \tfrac{1}{2} \times 10 \times 14.142 \times 0.9659 \approx 68.30$

$R = \frac{10}{2 \sin 30°} = \frac{10}{2 \times 0.5} = 10$

The circumradius of 10 means the triangle fits inside a circle of radius 10. Note that $b/\sin B = 14.142/\sin 45° = 14.142/0.7071 = 20 = 2R$ — confirming the identity.

Derivation from the circumscribed circle

Draw the circumscribed circle of radius $R$ around the triangle. Drop a diameter from vertex $A$ through the center to a point $D$ on the circle. Because $\angle ABD$ is inscribed in a semicircle, it is a right angle, so $\sin(\angle ADB) = a/(2R)$.

The Inscribed Angle Theorem tells us that $\angle ADB = A$ (both angles subtend arc $BC$). Therefore:

$\sin A = \frac{a}{2R} \implies \frac{a}{\sin A} = 2R$

The same argument applies symmetrically to sides $b$ and $c$, giving the full Law of Sines ratio.

Law of Sines vs. Law of Cosines

A simple test distinguishes the two: if a known angle pairs directly with the known side opposite it, the Law of Sines applies; if the known angle sits between two known sides, the Law of Cosines applies.

The ambiguous case

The ambiguous case arises only in the SSA configuration (two sides + a non-included angle). When side $a < b \sin A$, no triangle exists. When $b \sin A < a < b$, two different triangles satisfy the same inputs — this is the ambiguity.

This calculator uses the AAS configuration, which is always unambiguous. Once two angles are specified, the third angle $C = 180° − A − B$ is forced, and the Law of Sines resolves uniquely. There is no second solution.