Three points which are unambiguously defining a triangle in the Cartesian system of coordinates are its tops. Knowing their situation it is relative each of coordinate axes it is possible to calculate any parameters of this flat figure, including also the area limited by its perimeter. It can be done in several ways.
Instruction
1. Use Heron's formula for calculation of the area of a triangle. The sizes of three parties of a figure are involved in it therefore fill calculations from their definition. Length of each party has to be equal to a root from the sum of squares of lengths of its projections to coordinate axes. If to designate coordinates of tops of A (X ₁, Y ₁, Z ₁), B (X ₂, Y ₂, Z ₂) and C (X ₃, Y ₃, Z ₃), lengths of their parties it is possible to express so: AB = √ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²), BC = √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²), AC = √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²).
2. For simplification of calculations enter an auxiliary variable - poluperimetr (P). From names i see that it is a half of the sum of lengths of all parties: Р = ½ * (AB+BC+AC) = ½ * (√ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²) + √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²) + √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²).
3. Calculate the area (S) by Heron's formula - take a root from the work of a poluperimetr on the difference between it and length of each of the parties. In a general view it can be written down so: S = √ (P * (PAB) * (P-BC) * (P-AC)) = √ (P * (P-√ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²)) * (P-√ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²)) * (P-√ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²)).
4. For practical calculations it is convenient to use specialized online calculators. These are the scripts placed on servers of some websites which will do all necessary calculations on the basis of the coordinates entered by you into the corresponding form. The only lack of such service - it does not offer explanations and justifications for each step of calculations. Therefore if you are interested only in the end result, but not calculations in a general view, pass, for example, to the http://planetcalc.ru/218/ page.
5. Into fields of a form separately enter each coordinate of each of triangle tops - they are designated as Ax, Ay, Az, etc. here. If the triangle is set by two-dimensional coordinates, to fields of z-coordinates - Az, Bz and Cz - write zero. In the field "Calculation Accuracy" determine the necessary number of signs after a comma, clicking with a mouse of the pictogram of plus or minus. It is not obligatory to press the orange Calculate button placed under a form, calculations will be made also without it. You will find the answer near the inscription "Area of a Triangle" - it is placed under the orange button at once.