Law of Sines
a/sin A = b/sin B = c/sin C.
About this topic
The Law of Sines relates sides to the sines of their opposite angles in any triangle: a/sin A = b/sin B = c/sin C. Use it when you know two angles and a side (AAS/ASA), or two sides and an angle not between them (SSA).
Each fraction equals the diameter of the triangle's circumscribed circle, which explains why the law holds. In practice, set up a proportion with one complete pair (a side and its opposite angle) and one unknown. When two angles are known, the third is 180° minus their sum.
The SSA case is ambiguous and can give zero, one or two triangles. If the side opposite the given angle is shorter than the height, no triangle exists; if it equals the height, one right triangle exists; if between the height and the other side, two triangles fit. Always check whether an obtuse supplement 180° − θ also works.
Example: with A = 40°, B = 70° and a = 8, we get b = 8 sin 70° / sin 40°, about 11.7. The law is used in surveying and navigation, to find distances across rivers or to ships. Tip: label each side with the same letter as its opposite angle, and keep calculator mode in degrees.
Key ideas
- Works in any triangle
- Use with AAS or ASA
- SSA may be ambiguous
Quick check
Use cases: where Law of Sines helps
- Homework: review the key idea behind Law of Sines before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Law of Sines without notes.
- Visual learning: play with the interactive lab to see how Law of Sines changes when you adjust its inputs.
- Teaching a friend: explain Law of Sines in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Law of Sines?
a/sin A = b/sin B = c/sin C.
What will I learn in Law of Sines?
Works in any triangle. Use with AAS or ASA. SSA may be ambiguous.
Is Law of Sines part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Law of Sines lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Law of Sines?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.