Scalene triangles are defined as a triangle where the interior angles are all different. Most triangles drawn at random would be scalene. The converse of this is also true - If all three angles are different, then the triangle is scalene, and all the sides are different lengths.

Area

The area of a scalene triangle is:

${\displaystyle A=\frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{4}}$

See the heron's formula for more information.

Perimeter

The perimeter of a scalene triangle is

${\displaystyle P=a+b+c}$

Trigonometric functions of half angles in a triangle

${\displaystyle \begin{align} \sin&=\sqrt{\frac{a^2-(b^2+c^2)}{4bc}+\frac12}\\ \cos&=\sqrt{\frac{(b^2+c^2)-a^2}{4bc}+\frac12}\\ \tan&=\sqrt{\frac{(a+b-c)(a-b+c)}{(a+b+c)(-a+b+c)}}\\ \csc&=2\sqrt{\frac{bc}{(a+b-c)(a-b+c)}}\\ \sec&=2\sqrt{\frac{bc}{(a+b+c)(-a+b+c)}}\\ \cot&=\sqrt{\frac{(a+b+c)(-a+b+c)}{(a+b-c)(a-b+c)}} \end{align}}$

Trigonometric functions of whole angles in a triangle

${\displaystyle \begin{align} \sin&=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{2A}{bc}=\frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{2bc}\\ \cos&=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b^2+c^2-a^2}{2bc}\\ \tan&=\frac{\text{opposite}}{\text{adjacent}}=\frac{4A}{b^2+c^2-a^2}=\sqrt{\frac{(a^2+b^2+c^2)^2- 2(a^4+b^4+c^4)}{(b^2+c^2-a^2)^2}}\\ \csc&=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{bc}{2A}=\frac{2bc}{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}\\ \sec&=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{2bc}{b^2+c^2-a^2}\\ \cot&=\frac{\text{adjacent}}{\text{opposite}}=\frac{b^2+c^2-a^2}{4A}=\sqrt{\frac{(b^2+c^2-a^2)^2}{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}\end{align}}$

CircumCircle

${\displaystyle R=\frac{abc}{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}}$

InCircle

${\displaystyle r=\frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{2(a+b+c)}}$