A worked example
A triangle with sides 3, 4, 5 — a classic right triangle — has angles of about 36.87°, 53.13° and exactly 90°, with an area of exactly 6.
Frequently asked questions
Why do the three side lengths need to satisfy a specific rule?
The triangle inequality theorem requires that any two sides together must be longer than the third — otherwise the three lengths simply can't connect into a closed triangle, no matter how you arrange them.
What's the Law of Cosines actually doing here?
It generalizes the Pythagorean theorem to work for any triangle, not just right triangles — given three sides, it solves for each angle; given two sides and the angle between them, it solves for the missing side.
What's the difference between inradius and circumradius?
The inradius is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. The circumradius is the radius of the circle that passes through all three vertices, exactly enclosing the triangle.