Guide
1
outcome in the calculation of that area triangle is half the product of the length of any of the parties to the length of the height dropped to this side.This definition implies that to find the area you need to know the height and length of the side pieces.
2
Begin by calculating the lengths of the sides of the triangle .Mark coordinates of the vertices of the figure as: A (X₁, Y₁, Z₁), B (X₂, Y₂, Z₂) and C (X₃, Y₃, Z₃).Then the length of side AB, you can calculate the formula AB = √ ((X₁-X₂) ² + (Y₁-Y₂) ² +
(Z₁-Z₂) ²).For the other two sides of the formula would look like: BC = √ ((X₂-X₃) ² + (Y₂-Y₃) ² + (Z₂-Z₃) ²) and AC = √ ((X₁-X₃) ² + (Y₁-Y₃) ² + (Z₁-Z₃) ²).For example, triangle with coordinates A (3,5,7), B (16,14,19) and C (1,2,13) ​​the length of the side AB 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 manner, will be equal to √ (15² + 12² + 6²) = √405 ≈ 20,12 and √ (2² + 3² + (-6²)) = √49 = 7.
3
Knowledge lengths of the three sides, obtained in the previous step, it is sufficient to calculate the area 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 triangle -sample, from the previous step, this formula gives a value: 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 * 3297) ≈ ¼ * √75768,55 ≈ ¼ * 275,26 = 68,815.
4
From Square triangle , calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the height of each of the parties.Since area is equal to half the product of the height to length of the side to which it is held, to find the height twice the area to divide the desired length of side: H = 2 * S / a.For example used above the height dropped to the side AB is 2 * 68,815 / 16,09 ≈ 8,55, to the height of the sun will have a length of 2 * 68,815 / 20,12 ≈ 6,84, and for the side speakers, this value will be equal to2 * 68.815 / 7 ≈ 19,66.