## Triangles

### Area of a Triangle

There are numerous ways to determine the area of a triangle. Each calculation method depends on what information is already known about the triangle in question.

Given the height and width of a right triangle, the area is simply:

$$A=\frac{1}{2}ab$$

Given the height and and base edge length of any triangle, the area is similarly:

$$A=\frac{1}{2}bh$$

Given one angle and is adjacent lengths, the area is:

$$A=\frac{1}{2}bc\,sin\,α$$

Given one side and its two adjacent angles, the area is:

$$A=\frac{a^2\,sin\,α\,sin\,β}{2\,sin(α+β)}$$

Given all three side lengths, we use Heron's formula (aka Hero's formula):

$$A=\sqrt{s(s-a)(s-b)(s-c)}$$

Where $$s$$ is the semiperimeter:

$$s=\frac{1}{2}(a+b+c)$$

Note that a circle inscribed within any triangle has the property:

$$r=\frac{A}{s}$$