Law of Sines and Cosines Calculator (Free Step-by-Step Triangle Solver)
Use this free law of sines and law of cosines calculator to solve any triangle. Choose what you already know — three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), or two sides and a non-included angle (the ambiguous SSA case) — and the tool finds every remaining side and angle, the area, and the perimeter, with a scaled diagram and a clear step-by-step solution.
When do you use the Law of Sines vs. the Law of Cosines?
Use the Law of Cosines when you know all three sides (SSS) or two sides and the angle between them (SAS) — these cases don’t give you a complete side-angle pair to start from. Use the Law of Sines when you already have a matching side and angle pair, which happens with two angles and a side (ASA or AAS) or with two sides and a non-included angle (SSA).
Law of Sines
In any triangle with sides a, b, c opposite angles A, B, C: a / sin A = b / sin B = c / sin C.
Law of Cosines
c² = a² + b² − 2ab·cos C (and the matching forms for a² and b²).
How to use the calculator
- Pick the case that matches what you know from the “I know” menu (SSS, SAS, ASA, AAS, or SSA).
- Enter the three known values. Sides are lengths; angles are in degrees.
- Press Solve to see all sides and angles, the area, the perimeter, a scaled triangle, and the steps.
The ambiguous case (SSA)
When you know two sides and an angle that is not between them, there can be zero, one, or two triangles that fit. The calculator detects this automatically and shows both possible triangles when they exist.
Looking for more tools? Browse all of our free math calculators and solvers, including the Triangle Calculator and the Trigonometry Calculator.
Frequently asked questions
Does this calculator use the Law of Sines or the Law of Cosines?
Both. It automatically chooses the right one for your case: the Law of Cosines for SSS and SAS, and the Law of Sines for ASA, AAS, and SSA. The steps show exactly which law was applied.
What is the ambiguous case (SSA)?
When you know two sides and a non-included angle, the data can fit zero, one, or two different triangles. The calculator checks for this and displays both valid triangles whenever two solutions exist.
Are the angles in degrees or radians?
All angles are entered and shown in degrees.
Does it show the area and perimeter?
Yes. Along with every side and angle, it reports the triangle’s area (using one-half the product of two sides and the sine of the included angle) and its perimeter.
Is the Law of Sines and Cosines calculator free?
Yes — it runs in your browser with no sign-up required.
How to use the Law of Sines and Cosines Calculator for homework
The Law of Sines and Cosines Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.
Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.
Before you enter the problem
- Rewrite the problem in a clean line so every value is easy to see.
- Use parentheses around grouped expressions, especially in fractions and exponents.
- Keep units with the numbers while you work, even if the calculator only asks for the numbers.
- Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
- Look for restrictions such as positive values only, a chosen interval, or a required domain.
How to read the result
After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.
Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.
Common mistakes to avoid
| Mistake | How to catch it |
|---|---|
| Typing the wrong sign | Compare each negative sign, subtraction symbol, and exponent with the original problem. |
| Rounding too early | Keep extra decimal places until the final step, then round only as directed. |
| Forgetting parentheses | Group numerators, denominators, powers, and multi-step expressions before calculating. |
| Ignoring units | Write the unit next to the final answer so the result has meaning. |
| Trusting an unreasonable result | Use estimation, a graph, or substitution to check whether the answer makes sense. |
Turn the answer into practice
One good way to study is to solve the problem by hand first, then use the Law of Sines and Cosines Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.
- Work the problem on paper and circle your final answer.
- Use the calculator to check the result.
- If the answers match, write one sentence explaining the method.
- If they do not match, compare each step until you find the first difference.
- Redo a similar problem without the tool to make sure the correction sticks.
When to use a calculator and when to work by hand
Use the Law of Sines and Cosines Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.
If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.
Related to This Article
More math articles
- How to Determine Functions?
- How to Prepare for FTCE General Knowledge Math Test?
- Algebra 1 Formula Flashcards (Free Visual Review)
- Free Colorado CMAS Grade 3 Math Practice: 49 Printable PDFs for Classrooms, Tutors, and Parents
- Georgia Grade 2 Math Worksheets — Free Printable PDFs
- How to Simplify Ratios? (+FREE Worksheet!)
- Estimation Practice — Rounding to Estimate Sums & Differences
- PERT Math FREE Sample Practice Questions
- 10 Most Common 5th Grade IAR Math Questions
- Full-Length 6th Grade PARCC Math Practice Test-Answers and Explanations
What people say about "Law of Sines and Cosines Calculator (Free Step-by-Step Triangle Solver) - Effortless Math"?
No one replied yet.