Versatile triangle is called such triangle which lengths of the parties are not equal among themselves. At the same time it is meant that no two parties are equal (otherwise the triangle would turn out isosceles). For calculation of the area of a versatile triangle several different formulas are used. All main options which can meet in practice and at the solution of geometrical tasks are considered.
It is required to you
- - calculator;
- - protractor;
- - ruler.
Instruction
1. To find the area of a triangle, increase length of its party by height (the perpendicular lowered on this party from opposite top) and divide the received work into two. In the form of a formula this rule looks as follows: S = ½ * and * h, where: S – the area of a triangle, and – length of its party, h – height, lowered on this party. Length of the party and height have to be presented in identical units of measure. At the same time the area of a triangle will turn out in the corresponding "square" units.
2. Example. On one of the parties of a versatile triangle 20 cm long, the perpendicular from opposite top 10 cm long is lowered. It is required to determine the area of a triangle. Decision.S = ½ * 20 * 10 = 100 (cm²).
3. If lengths of two any parties of a versatile triangle and a corner between them are known, then use a formula: S = ½ * and * b * sinγ, where: and, b are lengths of two any parties, and γ – corner size between them.
4. In practice, for example, at measurement of land area, use of the above-stated formulas sometimes can be difficult as demands additional constructions and measurement of corners. If lengths of all three parties of a versatile triangle are known to you, then use Heron's formula: S = √ (p(p-a) (p-b) (p-c)), where: a, b, c are lengths of the parties of a triangle, r – poluperimetr: p = (a+b+c)/2.
5. If except lengths of all parties the radius of an inscribed in a triangle circle is known, then use the following compact formula: S = p * r, where: r – radius of an inscribed circle (r – poluperimetr).
6. For calculation of the area of a versatile triangle through the radius of a circumscribed circle and length of its parties, use a formula: S = ABC/4R, where: R – radius of a circumscribed circle.
7. If length of one of the parties of a triangle and size of three corners (in principle, it is enough of two – the size of the third is calculated from equality of the sum of three corners of a triangle - 180º), then use a formula is known: S = (a² * sinβ * sinγ)/2sinα, where α – size opposite to the party and a corner; β, γ – sizes of other two corners of a triangle.