Finding out the area of a triangle if the coordinates of the three vertices are given

What you are looking for is called the shoelace formula:

$$\begin{align*} \text{Area} &= \frac12 \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|\\ &= \frac12 \big| x_A y_B + x_B y_C + x_C y_A - x_A y_C - x_C y_B - x_B y_A \big|\\ &= \frac12 \Big|\det \begin{bmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{bmatrix}\Big| \end{align*}$$

The last version tells you how to generalize the formula to higher dimensions.

PS. Another generalization of the formula is obtained by noting that it follows from a discrete version of the Green’s theorem:

$$\text{Area} = \iint_{\text{domain}}dx\,dy = \frac12\oint_{\text{boundary}}x\,dy - y\,dx$$

Thus the signed (oriented) area of a polygon with $n$ vertexes $(x_i,y_i)$ is

$$\frac12\sum_{i=0}^{n-1} x_i y_{i+1} - x_{i+1} y_i$$

where indexes are added modulo $n$.