About this calculator
The Law of Sines Calculator applies the sine rule to find missing sides and angles in a non-right triangle. It is useful when you know an angle-side opposite pair plus another angle or side, especially in ASA, AAS, and some SSA problems.
law of sines calculator method
The sine rule says each side divided by the sine of its opposite angle gives the same ratio. Once one opposite side-angle pair is known, the ratio can be used to find other sides or angles.
- a / sin(A) = b / sin(B) = c / sin(C)
- side = common ratio x sin(opposite angle)
- angle = sin^-1(side / common ratio)
How to use the law of sines calculator
- Enter the known sides and angles.
- Make sure at least one side is paired with its opposite angle.
- If two angles are known, calculate the third angle from 180 degrees.
- Find the common sine-rule ratio.
- Use the ratio to calculate missing sides.
- Use inverse sine to calculate missing angles where appropriate.
- Check for possible ambiguous SSA cases.
Worked examples
Two angles and one side
Input: A = 40 degrees, B = 65 degrees, a = 7
Calculation: C = 75 degrees, common ratio = 7 / sin(40)
Result: b and c found from the common ratio
Side-opposite-angle pair
Input: A = 30 degrees, a = 5, b = 8
Calculation: B = sin^-1(8 x sin(30) / 5)
Result: B about 53.13 degrees for the principal solution
Ambiguous SSA case
When two sides and a non-included angle are known, the sine rule can sometimes produce two possible triangles. The calculator gives the principal solution, so draw the triangle or use the general triangle calculator if ambiguity matters.
Common mistakes to avoid
- Using the wrong side label
- Triangle formulas depend on matching sides with their opposite angles or with the correct right-triangle role. If a result looks impossible, recheck the labels before changing the formula.
- Forgetting triangle validity
- Not every set of side lengths can form a triangle. The longest side must be shorter than the sum of the other two sides.
- Rounding too early
- Keep extra decimal places while calculating, especially when using square roots, sine, cosine, or inverse trig. Round the final answer to the precision required.
Edge cases
- A right triangle must have one angle of exactly 90 degrees.
- The hypotenuse must be the longest side in a right triangle.
- Special triangle ratios only apply to exact 30-60-90 and 45-45-90 triangles.
- The sine rule can produce an ambiguous SSA case where two triangles are possible.
Limitations
This calculator is for educational maths support. It uses standard geometry and trigonometry formulas with decimal approximations. For exams, coursework, engineering, surveying, or construction, follow the required method, units, tolerances, and checking process.
Frequently asked questions
Which triangle calculator should I use?
Use Pythagoras or the right triangle calculator for right-angled triangles, special triangle calculators for exact 30-60-90 or 45-45-90 triangles, and the sine or cosine law calculators for non-right triangles.
What do sides a, b, and c mean?
In general triangle notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Some right-triangle pages also label c as the hypotenuse.
Why do triangle angles add to 180 degrees?
In ordinary flat Euclidean geometry, the interior angles of a triangle always add to 180 degrees. That rule is used throughout these calculators.
Can the calculator handle any units?
Yes for lengths, as long as all side inputs use the same unit. Area results are in square units based on the unit entered.
Why is my result impossible?
The inputs may not form a valid triangle, the hypotenuse may not be the longest side, or a side may have been paired with the wrong angle.
Related calculators
- Law of Cosines Calculator
- Triangle Calculator
- Trigonometry Calculator