# How do you find the area of a triangle in a 3D graph?

How do you find the area of a triangle in a 3 dimensional graph? Is it any different than a regular 2d graph? How would you solve it, if these were your three points? A(1,-4,-2), B(3,-3,-3), C(5,-1,-2)? Thanks for any help!

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The area of a triangle is always $bh\over 2$. If it is actually a spherical sector or some non-plane bounded by a triangle, the area changes. So you only need to find the lengths of the sides and some way of identifying the base and the height. –  abiessu Sep 28 '13 at 2:41

If you know what a vector cross product is, just take half of the magnitude of the cross product of the vectors $AB$ and $AC$.

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So solution is

$\text{Area} = 2.693$

$\text{Sides}: a = 3,\ b = 5,\ c = 2.449$

Using $3D$ triangle calculator: http://www.triangle-calculator.com/?what=vc&a=1&a1=-4&a2=-2&b=3&b1=-3&b2=-3&c=5&c1=-1&c2=-2&submit=Solve&3d=1

Answer by Ross Millikan is the best algorithm to solve this.

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Heron's formula gives you the area of a triangle from the length of the sides. Let the sides be $a,b,c$ and $s=\frac{a+b+c}2$ then the area $A=\sqrt{s(s-a)(s-b)(s-c)}$. Works in any dimension.

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Let $M$ be the matrix with column vectors $C-A$ and $C -B$. Then $|\det(M^tM)|$ is the area of the parallelogram spanned by the column vectors of $M$. Works in any dimension.

Michael

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