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. The term can more widely be used for triangles without any symmetries (other than the identity).

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.