Triangle – the simplest mathematical polygon consisting of three tops and the parties. The main quantity characteristic of a triangle, the area, is calculated in several ways on the basis of various measurements: lengths of the parties and height, corners between the parties, perimeter, radiuses of the entered and circumscribed circle and so forth.
Instruction
1. The basic formula of the area of any triangle of ABC is calculated as follows: S =? *c*h, where with – the basis of a triangle, h – height which is carried out to this basis.
2. Formula of calculation of the area through the work of the parties and sin of a corner between them: S =? *a*b*sin?.
3. Let r radius circle be entered in a triangle, then the formula of the area of a triangle will have an appearance: S =? *P*r, where P – triangle perimeter, i.e. S =? * (a + b + c) *r.
4. Let around a triangle R radius circle be described. A formula of the area of a triangle through the radius of a circumscribed circle and length of the parties of a triangle: S = (a*b*c)/(4*R). A formula of the area of a triangle through the radius of a circumscribed circle and corners of a triangle: S = 2*R^2*sin? *sin? *sin?.
5. There is Heron's formula for the area of a triangle called by name the Ancient Greek mathematician Heron Alexandriyski living in the beginning of our era. This formula gives definition to the area through lengths of all parties of a triangle: S =? to *v ((a + b + c) * (b + with - a) * (a + with - b) * (a + b - c)). Record of a formula with introduction of a concept of a poluperimetr becomes simpler: S = v (p * (p - a) * (p - b) * (p - c)), where p = (a + b + c)/2 - poluperimetr.
6. Formula of the area of a triangle through length of the party and corners of a triangle: S = a^2*sin? *sin? / (2*sin?), where? and? – adjacent corners, and? – opposite corner to the party of a.
7. For a rectangular triangle the formula of the area becomes simpler and looks as follows: S =? *a*b, i.e. the area of a rectangular triangle is equal to a half of the work of lengths of legs.
8. Area formula for an equilateral triangle: S = (a^2*v3)/4.
9. Area formula for an isosceles rectangular triangle: S =? * (a^2 + b^2), where an and b – triangle legs. Besides, for any triangle fairly following inequality: S