Law of Sines Calculator
Enter a side with its opposite angle, then one more measurement, and this calculator applies the law of sines to find the missing part. It solves for a missing side or angle, then completes the triangle by computing area, perimeter, inscribed-circle radius, and circumscribed-circle radius. The ambiguous SSA case is detected and both solutions shown.
Formula
Worked example
Given a = 8 m with opposite angle A = 40°, and second angle B = 60°: ratio = 8 / sin 40° = 12.4511. Side b = 12.4511 * sin 60° = 10.7827 m. Angle C = 180 - 40 - 60 = 80°; side c = 12.4511 * sin 80° = 12.2629 m. Area by Heron: s = (8 + 10.7827 + 12.2629) / 2 = 15.5228; K = sqrt(15.5228 * 7.5228 * 4.7401 * 3.2599) = 46.65 m². Inradius r = 46.65 / 15.5228 = 3.0051 m; circumradius R = ratio / 2 = 6.2256 m.
What the law of sines actually says
In any triangle the ratio of each side to the sine of the angle directly opposite it is the same constant. Written out: a divided by sin A equals b divided by sin B equals c divided by sin C. This common ratio also equals the diameter of the circumscribed circle (2R), which links the law of sines directly to circle geometry. As soon as you know one matched side-angle pair, that ratio is fixed, and any other angle you know immediately yields its opposite side (or vice versa). The law applies to every triangle, not just right triangles, which is what makes it essential in surveying, navigation, and physics.
When to reach for it, and when not to
Reach for the law of sines when your data includes at least one side together with its opposite angle. That covers AAS (two angles plus a non-included side), ASA (two angles plus the included side, after deriving the third angle by subtracting from 180°), and SSA (two sides and a non-included angle). It cannot start from SSS or SAS, because neither gives you a complete side-angle pair. In those cases start with the law of cosines to unlock the first pair, then finish with the law of sines. If you are working in radians rather than degrees, toggle the angle unit and all inputs and outputs update automatically.
The ambiguous SSA case explained
When you know two sides and a non-included angle, the arcsine only returns the acute angle, yet its supplement (180° minus that value) can also satisfy the equation. Before deciding, check the altitude: h = b * sin A. If the known opposite side a is shorter than h, no triangle exists. If a equals h, exactly one right triangle exists. If a is between h and b, two triangles are possible and both solutions are shown. If a is at least as long as b, the obtuse solution forces the angles to exceed 180° so only the acute answer is valid.
Area, perimeter, and the two special radii
Once all three sides are known, Heron's formula gives the area: K = sqrt(s * (s-a) * (s-b) * (s-c)), where s is the semi-perimeter (a+b+c)/2. The inscribed circle (incircle) fits snugly inside the triangle; its radius is r = K / s. The circumscribed circle (circumcircle) passes through all three vertices; its radius is R = abc / (4K), and note that it also equals the common law-of-sines ratio divided by two. These radii are particularly useful in architecture, cartography, and gear design where the enclosing or inscribed circle matters.
Triangle cases: which law applies, and what to expect
| Known parts | Case name | Best first law | Ambiguous? | Outputs here |
|---|---|---|---|---|
| Two angles + non-included side | AAS | Law of sines | No | All sides, angles, area, r, R |
| Two angles + included side | ASA | Law of sines | No | All sides, angles, area, r, R |
| Two sides + non-included angle | SSA | Law of sines | Sometimes (check h) | Both solutions if ambiguous |
| Two sides + included angle | SAS | Law of cosines first | No | Use law-of-cosines calculator |
| Three sides | SSS | Law of cosines first | No | Use law-of-cosines calculator |
| Three angles only | AAA | Neither alone | Yes (infinite) | Similar triangles, not unique |
Use this table to pick the right tool before you start entering numbers.
Frequently asked questions
Can I enter angles in radians?
Yes. Toggle "Angle unit" to Radians at the top of the form. All angle inputs and outputs will then be in radians. The math is identical; the calculator multiplies by 180/pi to convert back for Heron's formula and circle-radius steps.
When can I use the law of sines instead of the law of cosines?
Use the law of sines whenever you know a side and the angle directly opposite it, plus one more side or angle (the AAS, ASA, or SSA cases). If you only know all three sides (SSS) or two sides and the angle between them (SAS), start with the law of cosines instead, then the law of sines can finish the job.
Why does my SSA problem have two answers?
That is the ambiguous SSA case. When you know two sides and an angle opposite one of them, the arcsine gives an acute angle, but its supplement (180° minus that value) can also form a valid triangle. This calculator checks the altitude h = b * sin A and reports both possibilities whenever the second one keeps all angles summing to less than 180°.
How is the circumscribed circle radius calculated?
The circumscribed circle (circumcircle) passes through all three vertices. Its radius is R = a / (2 sin A), which is simply half the common law-of-sines ratio. The formula R = abc / (4K) using Heron's area K gives the same result and is used here as a cross-check.
What does the inscribed circle radius tell me?
The incircle is the largest circle that fits entirely inside the triangle, touching all three sides. Its radius is r = K / s, where K is the area and s is the semi-perimeter. It is useful in engineering contexts where a rounded corner or fitting must stay within the triangle.
What length units does this calculator support?
You can enter side lengths in metres, centimetres, millimetres, kilometres, feet, inches, yards, or miles. All three sides are converted to metres internally to ensure consistent area and radius outputs. You pick the unit alongside the number, and all outputs (including area, which is in unit-squared) match that choice.