How to find the area of a triangle by three points

Use Heron's formula to calculate the area of a triangle. It involves the dimensions of three faces, so the calculations start with their definition. The length of each side must be equal to the square root of the sum of the squares of the lengths of its projections on the coordinate axes. If the coordinates of the vertices are A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃), the lengths of their sides can be expressed as: AB = √((X₁-X₂)2 + (Y₁-Y₂)2 + (Z₁-Z₂)2), BC = √((X₂-X₃)2 + (Y₂-Y₃)2 + (Z₂-Z₃)2) AC = √((X₁-X₃)2 + (Y₁-Y₃)2 + (Z₁-Z₃)2).

To simplify the calculations, enter the auxiliary variable properiter (R). As the name suggests, which is half the sum of the lengths of all sides: P = ½*(AB+BC+AC) = ½*(√((X₁-X₂)2 + (Y₁-Y₂)2 + (Z₁-Z₂)2) + √((X₂-X₃)2 + (Y₂-Y₃)2 + (Z₂-Z₃)2) + √((X₁-X₃)2 + (Y₁-Y₃)2 + (Z₁-Z₃)2).

Calculate the area (S) by Heron's formula - extract the root of the product of properiety on the difference between him and the length of each of the parties. In General it can be written as: S = √(P*(P-AB)*(P-BC)*(P-AC)) = √( P*(P-√((X₁-X₂)2 + (Y₁-Y₂)2 + (Z₁-Z₂)2))*(P-√((X₂-X₃)2 + (Y₂-Y₃)2 + (Z₂-Z₃)2))*(P-√((X₁-X₃)2 + (Y₁-Y₃)2 + (Z₁-Z₃)2)).

For practical calculations it is convenient to use specialized online calculators. These are the scripts that are hosted on servers of some sites that will do all the necessary calculations based on the coordinates you entered in the appropriate form. The only disadvantage of this service - it gives no explanations and justifications for each step of the calculation. Therefore, if you are only interested in the end result, not the calculation in a General view, go, for example, on the page http://planetcalc.ru/218/.

In the form fields separately enter each coordinate of each vertex of the triangle - they are denoted here as Ax, Ay, Az, etc. If a triangle is specified two-dimensional coordinates, in the field of applicator that con - Az, Bz and Cz - write zero. In the "calculation Precision" set the desired number of decimal places by clicking the mouse icon to a plus or minus. Placed under the form the orange "Calculate" button to press is not necessary, the calculations will be made without it. You will find the answer next to "Square the triangle" - it is placed just under the orange button.