Law of Sines Calculator

Example data

Formula used

For any triangle, the law of sines links each side to its opposite angle: a / sin(A) = b / sin(B) = c / sin(C) = 2R, where R is the circumradius.

• k = a / sin(A) is a shared ratio used to find unknown sides.
• If two angles are known, the third is 180° − (sum of known angles).
• For SSA inputs, sin(B) = b / k can produce two angles.
• When needed, the law of cosines is used to compute a missing opposite side in SAS/SSS cases.

How to use this calculator

1. Enter known angles and sides using the A↔a, B↔b, C↔c pairing.
2. Leave unknown fields blank; do not type question marks.
3. Select your angle unit (degrees or radians).
4. Press Submit to see results above the form.
5. If two solutions appear, your data is an SSA case.
6. Use the export buttons to save the result table.

When the law of sines is the right tool

Use this method when a triangle provides at least one opposite pair, such as angle A with side a, plus one more angle or side. It excels in AAS and ASA setups because the remaining angle is fixed by the 180° sum. In SSS and SAS situations, this calculator can still produce results by first applying the cosine relationship to establish a consistent triangle, then returning to sine ratios.

Understanding the shared ratio k

The calculator forms k = a/sin(A), which equals b/sin(B) and c/sin(C). Once k is known, any missing side is found by side = k·sin(opposite angle). This ratio also equals 2R, where R is the circumradius. Reporting R helps compare triangles with similar shapes because R scales linearly with every side length.

Handling the ambiguous SSA case

SSA inputs can produce two valid triangles because arcsin returns an acute angle, yet an obtuse supplement may share the same sine value. The calculator tests both candidates, completes the third angle, and verifies consistency with your provided sides. If two solutions appear, review which triangle matches your diagram, context, or measurement constraints.

Quality checks that protect your results

Angles must be positive and their sum must equal 180°. Sides must be positive, and for three-side entries the triangle inequality must hold. Internally, trigonometric values are clamped to prevent rounding errors from producing impossible acos or asin calls. These checks reduce false solutions when inputs are near-degenerate or heavily rounded.

Interpreting the table, area, and heights

Each solution lists angles, sides, area, and circumradius. Area is computed with Heron’s expression, which uses only side lengths, and heights follow from h = 2·Area/side. This lets you cross-check geometry: a larger side should typically correspond to a larger opposite angle, and heights should shrink as their opposite sides grow.

Using exports and charts for reporting

The CSV export is ideal for spreadsheets and classroom datasets, while the PDF export creates a shareable summary with inputs and computed values. Plotly charts visualize side and angle magnitudes immediately, helping spot entry mistakes like swapped labels. For documentation, record the unit choice, the chosen solution number in SSA cases, and any rounding applied to measurements securely.

FAQs

1) Why do I sometimes get two solutions?

That happens in SSA inputs. A side and a non-included angle can form an acute or obtuse triangle with the same sine value. The calculator tests both and lists valid solutions.

2) Do I need to enter all three angles?

No. If you enter two angles, the third is computed from the 180° rule. Pair at least one angle with its opposite side to scale the triangle.

3) Why are results shown in degrees even if I select radians?

Degrees are easier to read for most geometry checks. If you select radians, inputs are converted internally, but the displayed angles remain in degrees for clarity.

4) What does k represent in the results?

k is the common ratio side/sin(opposite angle). It is identical for all three pairs in a valid triangle. It also equals 2R, linking directly to the circumradius.

5) Can I solve a triangle using only sides?

Yes. If you provide a, b, and c, the calculator computes angles using the cosine relationship, checks the triangle inequality, then reports the full solution set.

6) Why might the calculator say no valid triangle exists?

Common causes are inconsistent measurements, angles that sum beyond 180°, or a side that is too long for the other two. Recheck labels, units, and rounding.