The Law of Sines Calculator finds missing angles and side lengths in a triangle using a single reliable rule. Enter two known values (like angle–side or angle–angle–side) and it computes the remaining sides and angles consistently.
This article explains the law, defines every variable, and shows practical examples so you can trust the calculator results and verify them quickly.
What the Law of Sines says
The Law of Sines links the ratio of each side to the sine of its opposite angle. It works for any triangle (acute, right, or obtuse), as long as you have enough information to solve it.
The formula
For a triangle with angles A, B, C and opposite sides a, b, c:
- a / sin(A) = b / sin(B) = c / sin(C)
Here, each side length is paired with the angle directly opposite it.
How to map triangle parts to the variables
To avoid mistakes, use this consistent labeling:
| Triangle element | Meaning | Opposite pair |
|---|---|---|
| Angle A | One interior angle | Opposite side a |
| Angle B | Another interior angle | Opposite side b |
| Angle C | The remaining interior angle | Opposite side c |
Angles in a triangle always satisfy A + B + C = 180°.
Common scenarios the calculator can solve
A Law of Sines Calculator typically handles multiple input patterns. The most common are:
- ASA/AAS: two angles and one side (often called “angle–angle–side”).
- AAS/ASA includes cases where a side is known and you compute the third angle from the angle sum.
- SSA: two sides and a non-included angle (for example, side a, side b, and angle A). This case can produce one solution or two solutions depending on the values.
When an SSA case allows two possible triangles, a calculator may show two angle values. The calculator below is designed to compute a consistent set of results and flag when the ambiguous case occurs.
What the calculator computes (outputs)
After you enter the known values, the calculator computes:
- Missing angles (A, B, and C) in degrees.
- Missing side lengths (a, b, and c) using the sine ratios.
It uses trigonometry in radians internally but displays angles in degrees by default.
Units and conversions (degrees vs. radians)
Angles are usually given in degrees. The calculator converts to radians internally because most math functions use radians.
- Angle input unit: choose degrees or radians.
- Angle output: shown in degrees for easy reading.
- Side input unit: treated as a plain length (m, cm, inches, etc.). The calculator keeps the same unit for outputs.
So if you enter side lengths in centimeters, the computed sides are also in centimeters.
Step-by-step: how to use the Law of Sines Calculator
- Choose your known inputs: enter two values that match a solvable pattern (commonly two angles and one side, or one angle and its opposite side plus another side/angle).
- Set units: pick degrees or radians for angles, and type side lengths in the unit you’re using.
- Click Calculate.
- Review results: confirm that the computed angles add up to 180° (within rounding).
If the inputs can’t form a real triangle, the calculator shows a clear error message.
Practical examples
Example 1: Two angles and one side (ASA)
Suppose you know A = 35°, B = 55°, and the side a = 12 cm (opposite angle A). First compute the third angle:
- C = 180° − 35° − 55° = 90°
Then use the Law of Sines:
- b = a · sin(B) / sin(A)
- c = a · sin(C) / sin(A)
The calculator performs these steps automatically and returns the full triangle.
Example 2: One angle and two sides (SSA)
Suppose A = 30°, a = 8, and b = 12. Because the known angle is opposite side a, you use:
- sin(B) = b · sin(A) / a
If sin(B) is between 0 and 1, there may be two possible B angles. A calculator can show both possibilities (the “ambiguous case”) or select the valid one based on triangle feasibility.
Common mistakes to avoid
- Mixing up angles and opposite sides: side a must oppose angle A, not any other angle.
- Using inconsistent units: if you choose radians for input, enter angles in radians (not degrees).
- Entering values that can’t form a triangle: SSA cases can lead to no real solution if the sine ratio is outside the valid range.
Frequently Asked Questions
How do you know which side is opposite which angle?
In standard triangle notation, side a lies directly across from angle A, side b across from angle B, and side c across from angle C. “Opposite” means the angle and side do not touch at the same vertex. If you label carefully, the Law of Sines works correctly.
Why does the SSA case sometimes have two solutions?
With SSA, you compute an angle from a sine value. Sine can match the same value at two different angles between 0° and 180°. If both angles create a valid triangle (with angles summing to 180°), you get two possible triangles.
What happens if the calculator says there is no valid triangle?
That message usually means the inputs contradict triangle geometry. For SSA, it can happen when the computed sine value is greater than 1 or less than 0, which is impossible for real angles. For SSS-like inputs, it may fail if angles cannot sum correctly.
Does the Law of Sines work for right triangles?
Yes. For a right triangle, one angle is 90°, so its opposite side uses sin(90°) = 1. The Law of Sines still applies and gives the same relationships you would get from the Pythagorean theorem and basic trig identities.
How accurate are the results?
The calculator computes using standard floating-point math and rounds to a reasonable number of decimal places. If you check that angles add to 180°, you may see tiny differences due to rounding. For most school and engineering tasks, this accuracy is more than sufficient.
Bottom line
The Law of Sines Calculator gives you a fast, consistent way to solve for missing triangle angles and sides. Use it with correct angle/side pairing, pick the right input units, and handle SSA carefully when two solutions are possible.
