Law of Sines Calculator

What is a law of sines calculator?

A law of sines calculator solves a triangle when you know one angle, the side directly across from it, and a second angle. From those three values it works out the third angle and the two missing sides. The law of sines is the relationship that ties the angles of any triangle to the lengths of the sides opposite them, so it works for acute, right, and obtuse triangles alike, not just right triangles.

In this calculator you enter angle $A$ in degrees, side $a$ (the side opposite angle $A$), and angle $B$ in degrees. It returns angle $C$, side $b$, and side $c$.

How does it work?

The law of sines states that the ratio of each side to the sine of its opposite angle is the same for all three sides of a triangle:

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

Because the interior angles of any triangle add up to $180^\circ$, the third angle follows immediately:

$C = 180^\circ - A - B$

Once every angle is known and one opposite side ($a$) is given, the remaining sides come straight from the ratios above:

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

For these formulas to describe a real triangle, both $A$ and $B$ must be positive and their sum must be less than $180^\circ$. If $A + B \ge 180^\circ$ there is no valid triangle, and the calculator leaves the results blank.

Worked examples

Example 1: a 30-60-90 triangle

Suppose $A = 30^\circ$, $a = 10$, and $B = 60^\circ$. First find the missing angle:

$C = 180^\circ - 30^\circ - 60^\circ = 90^\circ$

Now apply the ratios. Since $\sin 30^\circ = 0.5$, $\sin 60^\circ \approx 0.8660$, and $\sin 90^\circ = 1$:

$b = \frac{10 \cdot \sin 60^\circ}{\sin 30^\circ} = \frac{10 \cdot 0.8660}{0.5} \approx 17.3205$

$c = \frac{10 \cdot \sin 90^\circ}{\sin 30^\circ} = \frac{10 \cdot 1}{0.5} = 20$

So $C = 90^\circ$, $b \approx 17.3205$, and $c = 20$.

Example 2: an isosceles right triangle

With $A = 45^\circ$, $a = 10$, and $B = 45^\circ$:

$C = 180^\circ - 45^\circ - 45^\circ = 90^\circ$

Because $\sin 45^\circ = \sin B$, side $b$ equals side $a$:

$b = \frac{10 \cdot \sin 45^\circ}{\sin 45^\circ} = 10$

$c = \frac{10 \cdot \sin 90^\circ}{\sin 45^\circ} = \frac{10}{0.7071} \approx 14.1421$

The triangle has two equal sides of length $10$ and a hypotenuse of about $14.1421$.

Practical notes

• Enter both angles in degrees. The calculator converts them internally before taking the sine.
• The known side $a$ must be the one opposite the known angle $A$; otherwise the ratios will not line up.
• This tool uses the angle-angle-side (AAS) configuration, which always produces a single triangle. The trickier side-side-angle (SSA) “ambiguous case” — where two different triangles can fit — is not handled here.
• When you instead know two sides and the angle between them, reach for the law of cosines calculator, and for plain sine, cosine, and tangent of a single angle see the trigonometry calculator.

Frequently asked questions

When should I use the law of sines instead of the law of cosines?

Use the law of sines when you know an angle together with the side opposite it, plus one more angle or side (the AAS or ASA cases). Use the law of cosines when you know two sides and the included angle, or all three sides.

Does the law of sines work for non-right triangles?

Yes. It applies to every triangle — acute, right, and obtuse. It is one of the main tools for solving triangles that are not right-angled.

Why are my results blank?

The results stay empty if a field is missing, if an angle is zero or negative, or if angle $A$ plus angle $B$ is $180^\circ$ or more, because no triangle can have those angles.