Geometry: Triangles

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:

\(r=\frac{A}{s}\)