Plane Geometry

Triangle

Perimeter, $P = a + b + c$

Semi-perimeter, $s = \frac{1}{2}P = \frac{1}{2}(a + b + c)$

Sum of included angles, $A + B + C = 180^\circ$

$\begin{align}
\text{Area, } A & = \frac{1}{2} bh \\
& = \frac{1}{2}ab \sin \theta \\
& = \sqrt{s(s - a)(s - b)(s - c)}
\end{align}$

Perimeter, $P = a + b + c + d$

Semi-perimeter, $s = \frac{1}{2}P = \frac{1}{2}(a + b + c + d)$

Sum of included angles, $A + B + C + D = 360^\circ$

$\begin{align}
\text{Area, } A & = \sqrt{(s - a)(s - b)(s - c)(s - d) - abcd \cos^2 \varphi} \\
& = \frac{1}{2} d_1 ~ d_2 \sin \theta
\end{align}$

where, $\varphi = \frac{1}{2}(A + C)$ or $\varphi = \frac{1}{2}(B + D)$

Perimeter, $P = nx$

Central angle of one segment, $\theta = \dfrac{360^\circ}{n}$

$\begin{align}
\text{Area, } A & = nA_1 \\
& = \dfrac{n}{2} xr \\
& = \dfrac{n}{2} R^2 \sin \theta
\end{align}$

Circumference, $c = 2\pi r = \pi d$

$\begin{align}
\text{Area, } A & = \pi r^2 \\
& = \dfrac{\pi}{4} d^2
\end{align}$