Law of Sines Calculator

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What Is the Law of Sines?

The Law of Sines is a trigonometry rule for solving triangles that are not necessarily right triangles. It connects each side of a triangle with the sine of the angle opposite that side.

In standard triangle notation, side \(a\) is opposite Angle \(A\), side \(b\) is opposite Angle \(B\), and side \(c\) is opposite Angle \(C\). The Law of Sines says that these matching side-and-angle ratios are equal:

This relationship is useful when a triangle has enough known information to connect at least one side with its opposite angle. It is commonly used for oblique triangles, which are triangles that are not right triangles.

The Law of Sines is especially helpful for:

• ASA: two angles and the included side
• SAA or AAS: two angles and a non-included side
• SSA: two sides and an angle that is not between them

The SSA case needs extra care because it can sometimes produce no triangle, one triangle, or two different triangles.

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Why the Law of Sines Matters

The Law of Sines gives students and problem-solvers a way to work with triangles when the Pythagorean theorem and basic right-triangle ratios are not enough. Many real triangle problems are oblique: a survey measurement, a slanted support, a navigation bearing, or a diagram in a trigonometry class may not include a right angle.

It also helps build an important habit in geometry and trigonometry: matching each side with the angle opposite it. A correct Law of Sines setup depends less on memorizing a procedure and more on reading the triangle labels accurately.

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Key Terms to Know

• Oblique triangle: A triangle that is not a right triangle.
• Opposite side: The side across from a given angle. Side \(a\) is opposite Angle \(A\).
• Included side: The side between two known angles. In ASA, the known side is included between the two known angles.
• Non-included side: A side that is not between the two known angles. In SAA or AAS, the known side is non-included.
• SSA: A side-side-angle situation where two sides and a non-included angle are known.
• Ambiguous case: An SSA situation where the same given information may describe two possible triangles, one triangle, or no triangle.
• Inverse sine: The operation used to find an angle from a sine value, often written as \(\sin^{-1}(x)\) or \(\arcsin(x)\).

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How the Law of Sines Works

A triangle's three interior angles always add to \(180^\circ\). When two angles are known, the third angle is found by subtraction:

Once one side and its opposite angle are known, the Law of Sines can scale the rest of the triangle. For example, if side \(c\) and Angle \(C\) are known, the shared ratio is:

Then the other sides can be found with:

This is why ASA and SAA triangles are usually direct to solve: two angles give the third angle, and the known side creates the scale of the triangle.

SSA is different. If \(a\), \(b\), and \(A\) are known, the Law of Sines gives:

Then an inverse sine gives a possible value of \(B\). But sine has the same positive value for an acute angle and its obtuse supplement. That means a second angle may also be possible:

Each possible value of \(B\) must still fit inside a triangle. The remaining angle is checked with:

Only solutions with positive angles are valid.

For SSA problems with an acute Angle \(A\), the altitude

is often used to understand the number of possible triangles:

For a right or obtuse Angle \(A\), the known opposite side \(a\) must be longer than \(b\) to form a valid triangle. Otherwise, no triangle is possible.

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Examples of the Law of Sines in Practice

Example 1: Solving an ASA Triangle

Suppose a triangle has:

• \(A = 60^\circ\)
• \(B = 45^\circ\)
• \(c = 10\)

First find the third angle:

$$ C = 180^\circ - 60^\circ - 45^\circ = 75^\circ $$

Now use side \(c\) and Angle \(C\) to create the shared ratio:

$$ k = \frac{10}{\sin 75^\circ} \approx 10.353 $$

Then solve for the missing sides:

$$ a = 10.353\sin 60^\circ \approx 8.966 $$

$$ b = 10.353\sin 45^\circ \approx 7.321 $$

So the solved triangle is approximately:

• \(A = 60^\circ\), \(B = 45^\circ\), \(C = 75^\circ\)
• \(a \approx 8.966\), \(b \approx 7.321\), \(c = 10\)

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Example 2: Solving an SAA Triangle

Suppose a triangle has:

• \(a = 8\)
• \(A = 35^\circ\)
• \(B = 65^\circ\)

Find the third angle:

$$ C = 180^\circ - 35^\circ - 65^\circ = 80^\circ $$

Use the known opposite pair \(a\) and \(A\):

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

Solve for \(b\):

$$ b = \frac{8\sin 65^\circ}{\sin 35^\circ} \approx 12.641 $$

Solve for \(c\):

$$ c = \frac{8\sin 80^\circ}{\sin 35^\circ} \approx 13.736 $$

The side lengths stay in the same unit as the entered side. If \(a = 8\) inches, then \(b\) and \(c\) are inches. If \(a = 8\) meters, then \(b\) and \(c\) are meters.

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Example 3: The Ambiguous SSA Case

Suppose a triangle has:

• \(a = 6\)
• \(b = 8\)
• \(A = 35^\circ\)

Use the Law of Sines to find a possible value of \(B\):

$$ \sin B = \frac{8\sin 35^\circ}{6} \approx 0.764769 $$

The first possible angle is:

$$ B_1 = \sin^{-1}(0.764769) \approx 49.886^\circ $$

The second possible angle is the supplement:

$$ B_2 = 180^\circ - 49.886^\circ \approx 130.114^\circ $$

Now check the remaining angle for each case.

For the first solution:

$$ C_1 = 180^\circ - 35^\circ - 49.886^\circ \approx 95.114^\circ $$

$$ c_1 = \frac{6\sin 95.114^\circ}{\sin 35^\circ} \approx 10.419 $$

For the second solution:

$$ C_2 = 180^\circ - 35^\circ - 130.114^\circ \approx 14.886^\circ $$

$$ c_2 = \frac{6\sin 14.886^\circ}{\sin 35^\circ} \approx 2.687 $$

Both remaining angles are positive, so both triangles are valid. This is why an SSA problem can have two solutions.

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How to Interpret the Result

The solved side lengths use the same length unit as the side values you entered. The calculator does not need to know whether the unit is inches, centimeters, feet, meters, or another unit, as long as every side input uses the same unit.

The angles are in degrees. For a valid triangle, the three angle results should add to \(180^\circ\), allowing for small differences caused by rounding.

A result showing two SSA solutions means the same side-side-angle information can form two different triangles. Neither solution is automatically “more correct” unless a diagram, context, or additional condition tells you which triangle is intended.

An em dash in a result field means the current inputs are incomplete or invalid. A triangle diagram is helpful for visualization, but the numeric side and angle results are the values to use for checking work.

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Common Mistakes and Misconceptions

Using radians instead of degrees: This calculator expects angle inputs in degrees. Entering radian values as if they were degrees will produce the wrong triangle or an invalid entry.

Mismatching sides and angles: Side \(a\) must be opposite Angle \(A\), side \(b\) must be opposite Angle \(B\), and side \(c\) must be opposite Angle \(C\). The Law of Sines only works when each side is paired with its opposite angle.

Mixing side units: Do not enter one side in inches and another side in centimeters. Use one consistent length unit for every side value.

Entering impossible angle sums: In ASA and SAA problems, the two entered angles must add to less than \(180^\circ\). If they add to \(180^\circ\) or more, there is no positive third angle left.

Assuming SSA always has one answer: SSA can produce zero, one, or two valid triangles. Always check whether the supplementary angle also creates a positive third angle.

Rounding too early: Rounding intermediate values can slightly change the final side lengths or angles. Keep more digits during hand calculations and round at the end.

Using the wrong mode: Choose the mode that matches the information you actually know. ASA, SAA, and SSA are different setups, even though they all may use the Law of Sines.

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When to Use the Law of Sines

Use the Law of Sines when you are solving an oblique triangle and you have enough information to form or derive a side-opposite-angle pair.

It is especially useful when:

• You know two angles and one side.
• You know two sides and an angle opposite one of those sides.
• You need to solve for all missing sides and angles of an ASA or SAA triangle.
• You need to analyze whether an SSA triangle has no solution, one solution, or two solutions.

If you are given all three sides, or two sides and the included angle, the Law of Cosines is usually the more direct starting point.

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Limitations and Things to Keep in Mind

The Law of Sines is a triangle-solving tool, not a guarantee that every set of inputs forms a triangle. Side lengths must be positive, angle measures must be positive and less than \(180^\circ\), and the final three angles must fit within \(180^\circ\).

This calculator supports ASA, SAA, and SSA setups. It does not solve every possible triangle input pattern. For example, it does not use SSS or SAS as direct modes, and it does not convert between length units.

Displayed side and angle results are rounded to a maximum of three decimal places. Intermediate sine values in the solution steps may show more precision. Because of rounding, checking the result by hand may give tiny differences in the last decimal place.

The triangle drawing is a visual aid. It is scaled to fit the display and should not be treated as an absolute measurement drawing.

For homework, teaching, and general trigonometry practice, calculator results are useful for checking and exploring solutions. For construction, engineering, surveying, navigation, or safety-related decisions, verify the calculation method, units, measurements, and rounding requirements before relying on the result.

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How to Use This Calculator

1. Choose ASA, SAA, or SSA mode.
2. Enter the required values for that mode:

• ASA: Angle \(A\), Angle \(B\), and side \(c\)
• SAA: side \(a\), Angle \(A\), and Angle \(B\)
• SSA: side \(a\), side \(b\), and Angle \(A\)

1. Use degrees for all angle inputs.
2. Use one consistent length unit for every side input.
3. Check the status message for incomplete or invalid entries.
4. Read the solved side lengths and angles.
5. For SSA ambiguous cases, switch between Solution 1 and Solution 2 to compare the two valid triangles.
6. Review the solution steps and triangle diagram. Use the graph download button only after a valid triangle has been rendered.

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Frequently Asked Questions

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Sources and References

Books and Open Textbooks

1. Jay Abramson. Precalculus 2e. OpenStax, 2021. Section 8.1, “Non-right Triangles: Law of Sines.” https://openstax.org/books/precalculus-2e/pages/8-1-non-right-triangles-law-of-sines
2. Michael Corral. Elementary Trigonometry. LibreTexts edition, Section 2.1, “The Law of Sines.” Last updated November 17, 2022. https://math.libretexts.org/Bookshelves/Precalculus/ElementaryTrigonometry%28Corral%29/02%3AGeneralTriangles/2.01%3ATheLawofSines
3. Carl Stitz and Jeff Zeager. Precalculus. LibreTexts edition, Section 11.2, “The Law of Sines.” Last updated October 3, 2022. https://math.libretexts.org/Bookshelves/Precalculus/Precalculus%28Stitz-Zeager%29/11%3AApplicationsofTrigonometry/11.02%3ATheLawofSines