Law of Sines Calculator

Understanding the Law of Sines

The Law of Sines (or sine rule) is an equation relating the lengths of the sides of any arbitrary triangle to the sines of its angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three pairs of sides and angles in a given triangle, and is equal to the diameter of the triangle's circumscribed circle (circumcircle).

For complementary triangle calculations, you can also use our Law of Cosines Calculator and Triangle Calculator.

The Law of Sines Formula

For any triangle with sides $a$, $b$, and $c$, and opposite angles $\alpha$, $\beta$, and $\gamma$, the relation is:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} = 2R$$

where $R$ represents the circumradius of the triangle.

Cases Solved by the Law of Sines

1. Angle-Angle-Side (AAS) & Angle-Side-Angle (ASA)

When any two angles are known (say $\alpha$ and $\beta$), the third interior angle is immediately determined by the triangle sum theorem:

$$\gamma = 180^\circ - \alpha - \beta$$

Once all three angles and at least one side length are established, the remaining side lengths are derived directly by cross-multiplying:

$$b = a \cdot \frac{\sin(\beta)}{\sin(\alpha)}, \quad c = a \cdot \frac{\sin(\gamma)}{\sin(\alpha)}$$

2. Side-Side-Angle (SSA): The Ambiguous Case

When two side lengths and an angle opposite to one of them are provided (e.g., $a$, $b$, and $\alpha$), the altitude from the opposite vertex is $h = b \sin(\alpha)$. Depending on the ratio of $a$ to $h$, there may be:

No triangle ($a < h$): Side $a$ is too short to reach the opposite baseline ($\sin(\beta) > 1$).
One right triangle ($a = h$): Side $a$ meets the base perpendicularly at $\beta = 90^\circ$.
Two distinct triangles ($h < a < b$): Side $a$ can swing inward or outward, creating an acute solution $\beta_1 = \arcsin(\frac{b \sin \alpha}{a})$ and an obtuse solution $\beta_2 = 180^\circ - \beta_1$.
One unique triangle ($a \ge b$): Only the acute angle solution fits the geometric requirement.

Worked Example: Solving an AAS Triangle

Suppose a triangle has $\alpha = 40^\circ$, $\beta = 60^\circ$, and side $a = 10\text{ cm}$.

Find the third angle: $\gamma = 180^\circ - 40^\circ - 60^\circ = 80^\circ$.
Calculate side $b$: $b = 10 \times \frac{\sin(60^\circ)}{\sin(40^\circ)} = 10 \times \frac{0.8660}{0.6428} \approx 13.47\text{ cm}$.
Calculate side $c$: $c = 10 \times \frac{\sin(80^\circ)}{\sin(40^\circ)} = 10 \times \frac{0.9848}{0.6428} \approx 15.32\text{ cm}$.
Compute triangle area: $\text{Area} = \frac{1}{2} a b \sin(\gamma) = 0.5 \times 10 \times 13.47 \times \sin(80^\circ) \approx 66.33\text{ cm}^2$.

Frequently Asked Questions

Why is SSA called the ambiguous case in trigonometry?

Because the sine function satisfies $\sin(\theta) = \sin(180^\circ - \theta)$, finding an angle via $\arcsin$ can yield both an acute angle and its supplementary obtuse angle, resulting in either zero, one, or two valid triangles.

Can the Law of Sines be used on right triangles?

Yes. For a right triangle where $\gamma = 90^\circ$, $\sin(90^\circ) = 1$, and the formula simplifies directly to standard definitions: $\sin(\alpha) = \frac{a}{c}$ and $\sin(\beta) = \frac{b}{c}$.

What is the geometric meaning of $2R$ in the Law of Sines?

$2R$ is the diameter of the circumcircle that passes through all three vertices of the triangle. Each side-to-sine ratio is identically equal to this circumdiameter.

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