Scalene triangle

The scalene triangle, or scalene, is a type of triangle with none of its three side lengths equal, and all of its three angles different.

In the context of a higher polytope, terminology is sometimes abused so that a triangle is called "scalene" if no symmetry of the polytope can transform one of the triangle's edges to another, such as in a snub cube, even when the triangle itself is isosceles or equilateral.

Measures
The area of a general triangle with side lengths a, b, and c satisfying the triangle inequality is given by Heron's formula:
 * $$A=\frac14\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}.$$

The law of sines allows one to then derive the following expression for the circumradius:
 * $$R=\frac{abc}{\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}}.$$

The angles α, β, and γ of the triangle, opposite to the sides with lengths a, b, c, respectively, are also given by the law of sines, as:
 * $$\alpha=\text{asin}\left(\frac{\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}}{bc}\right),$$
 * $$\beta=\text{asin}\left(\frac{\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}}{ca}\right),$$
 * $$\gamma=\text{asin}\left(\frac{\sqrt{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}}{ab}\right).$$

In vertex figures
Scalene triangles occur as vertex figures of 7 omnitruncated polyhedra.