How do I find the area of a plane, whose normal vector is given by $$ a\overrightarrow{i}+b\overrightarrow{j}+c\overrightarrow{k} $$ a,b and c are Real numbers.
I would have four 3-D coordinates which form this plane, and using these coordinates I will have to find out the area of the plane formed by the planes. I am writing code for this and will be required to perform this operation multiple times, so I am not inclined towards using the cross product
The initial idea that I had was to perform a series of rotations to transform the plane parallel to one of the XY, YZ or ZX planes. But I am not able to figure out the axes about which these have to be performed.
Assuming you have 4 completely different 3D points, graphically the plane looks something like this:
As seen in the image above the, the area is equal to $b \times h$, where $b$ = length between $C_2 \text{ and } C_3$, and $d$ = length between $C_3 \text{ and } C_4$.
For more complex shapes (shape of the plane), split them out into the basic shapes first, then continue with the method above. (Note: For triangles, Area = $\frac{1}{2} \times \text{ base } \times \text{ height }$):
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Hope this helps!