Determining the Degrees of an Angle Given Three X and Y Coordinates

[xyle]

How do I determine the degrees of an angle if I three X and Y coordinates? I honestly just need a formula to plug into some code. Thank you in advance.

 

[MarkFL]

How do I determine the degrees of an angle if I three X and Y coordinates? I honestly just need a formula to plug into some code. Thank you in advance.

Suppose the 3 points are given by:

\(\displaystyle \left(x_i,y_i\right)\) where \(\displaystyle i\in\{1,2,3\}\)

Now further suppose we wish to make one line segment from point 1 to point 2, and another from point 2 to point 3, and then find the angle, in degrees, subtended by the two segments. I would begin by defining the vectors:

\(\displaystyle a=\left\langle x_2-x_1,y_2-y_1 \right\rangle\)

\(\displaystyle b=\left\langle x_3-x_2,y_3-y_2 \right\rangle\)

And then, from the dot product of the two vectors, we may write:

\(\displaystyle \theta=\frac{180^{\circ}}{\pi}\arccos\left(\frac{a\cdot b}{|a||b|}\right)\)

where:

\(\displaystyle a\cdot b=\left(x_2-x_1\right)\left(x_3-x_2\right)+\left(y_2-y_1\right)\left(y_3-y_2\right)\)

\(\displaystyle |a|=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

\(\displaystyle |b|=\sqrt{\left(x_3-x_2\right)^2+\left(y_3-y_2\right)^2}\)

Does that make sense?