You are watching: Area of a triangle in 3d
If your 3 points are A, B, C then you may use directly the (half) overcome product formula :$$S=dfracinvernessgangshow.netbfAB imesinvernessgangshow.netbfAC2=dfrac2 $$that is (see the Wikipedia connect to get the cross-product in $invernessgangshow.netbbR^3$) :$$S=frac 12 sqrt(y_ABcdot z_AC-z_ABcdot y_AC)^2+(z_ABcdot x_AC-x_ABcdot z_AC)^2+(x_ABcdot y_AC-y_ABcdot x_AC)^2$$if $invernessgangshow.netbfAB=(x_AB,y_AB,z_AB)$ and $invernessgangshow.netbfAC=(x_AC,y_AC,z_AC)$
Say you have 3 points $invernessgangshow.netbfA, B, C$. Find the edge $ heta$ between $invernessgangshow.netbfAB$ and $invernessgangshow.netbfAC$ making use of dot product (i.e. $invernessgangshow.netbfABcdotinvernessgangshow.netbfAC=|invernessgangshow.netbfAB||invernessgangshow.netbfAC|cos heta$) and then girlfriend can find the area that the triangle using $$A=frac12|invernessgangshow.netbfAB||invernessgangshow.netbfAC|sin heta$$
Probably among the ideal ways (and possibly the very least computationally intensive ways) to approach this difficulty is with vectors. In this instance we have three points, i will keep them together arbitrary variables for far better reproducibility.
$$P_1(a_1,a_2,a_3); P_2(b_1,b_2,b_3); P_3(c_1,c_2,c_3)$$
These 3 points have the right to be offered to develop two vectors which share the exact same initial point. That does not matter in what combination we select the points, so long as we develop two vectors with the exact same initial allude to then calculation their normal (orthogonal) vector utilizing the cross product. As soon as we have actually the orthogonal, we can acquire its magnitude which will certainly equate to 2 times the area of the claimed triangle. (The size of the orthogonal will provide us the area of a parallelogram sharing the same nearby sides, as such we will half this area in the finish to gain the area of the triangle).
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To produce the vectors native a pair that points, you do the following:
$$ vecP_1P_2 = = $$$$ vecP_1P_3 = = $$
Here is a an easy picture of what we will be doing:
Once you have actually both vectors you deserve to calculate your orthogonal vector by taking their cross product, you carry out it together follows:
$$vecu = vecP_1P_2 imes vecP_1P_3$$$$vecu = eginvmatrix invernessgangshow.netbf ns & invernessgangshow.netbf j & invernessgangshow.netbf k \ x_1 & y_1 & z_1 \ x_2 & y_2 & z_2 \endvmatrix\$$
$$vecu = eginvmatrixy_1 & z_1\y_2 & z_2endvmatrixinvernessgangshow.netbf ns -eginvmatrixx_1 & z_1\x_2 & z_2endvmatrixinvernessgangshow.netbf j +eginvmatrixx_1 & y_1\x_2 & y_2endvmatrixinvernessgangshow.netbf k$$
This boils down to:
$$vecu = (y_1z_2-y_2z_1)invernessgangshow.netbf i-(x_1z_2-x_2z_1)invernessgangshow.netbf j+(x_1y_2-x_2y_1)invernessgangshow.netbf k\= \= $$
Once you have actually the orthogonal vector from the cross product, us calculate that magnitude:
$$|vecu| = sqrt(x_3)^2+(y_3)^2+(z_3)^2 = invernessgangshow.netbfArea of Parallelogram$$
Finally we room left v the area that a parallelogram written of two of our triangles. In ~ this suggest we fifty percent the magnitude, and also we will have the preferred result.