Assume in the calculations that the area of a triangle is equal to half of a work the length of either of the parties on the length of the height lowered on this side. From this definition it follows that in order to find the height you need to know the area of the shape and the length of the side.
Start with calculation of the lengths of the sides of the triangle. Label the coordinates of the vertices: A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃). Then the length of the side AB, you can calculate by the formula AB = √((X₁-X₂)2 + (Y₁-Y₂)2 + (Z₁-Z₂)2). For the other two sides of these formulas are as follows: BC = √((X₂-X₃)2 + (Y₂-Y₃)2 + (Z₂-Z₃)2) and AC = √((X₁-X₃)2 + (Y₁-Y₃)2 + (Z₁-Z₃)2). For example, for a triangle with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) the length of the side AB will be √((3-16)2 + (5-14)2 + (7-19)2) = √(-132 + (-92) + (-122)) = √(169 + 81 + 144) = √394 ≈ 19,85. The lengths of the sides BC and AC, calculated in the same way, will be equal √(152 + 122 + 62) = √405 ≈ 20,12 and √(22 + 32 + (-62)) = √49 = 7.
Knowledge of the lengths of the three sides, obtained in the previous step, it is enough to compute the area of the triangle (S) by Heron's formula: S = ¼ * √((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). For example, after substituting in this formula the values obtained from the coordinates of the trianglea sample from the previous step, this formula will give a value of S = ¼ *√((19,85+20,12+7) * (20,12+7-19,85) * (19,85+7-20,12) * (19,85+20,12-7)) = ¼ *√(46,97 * 7,27 * 6,73 * 32,97) ≈ ¼ *√75768,55 ≈ ¼*275,26 = 68,815.
Based on the area of the trianglecalculated in the previous step, and the lengths of the sides obtained in the second step, calculate the height for each of the parties. Since the area is equal to half the height works for the length of the side to which it is held, to find the height divide the double area for length right side: H = 2*S/a. For of the example used above the height lowered on the side AB will be 2*68,815/16,09 ≈ 8 and 55, the height to the side of the sun will have a length of 2*68,815/20,12 if 6,84, but for AC, this value will be equal to 2*68,815/7 if of 19.66.