If all points in perimeter of a circle do not exceed the limit of perimeter of a triangle and at the same time the perimeter of a circle has on only one general point from each of the parties of a triangle, then the circle is called inscribed in a triangle. There is only one value of radius of a circle at which it can be entered in a triangle with the set parameters. This property of the entered circle allows to calculate in parameters of a triangle and its parameters, including circle length.
Instruction
1. Begin calculation of length of an inscribed in a triangle circle (l) with determination of its radius (r). If the area of a polygon (S) and length of all its parties is known (a, b and c), radius will be equal to the relation of the doubled area to the sum of these lengths of r=2*S / (a+b+c).
2. Use geometrical definition of a constant of Pi for calculation of length of a circle on the known value of radius. This constant expresses the circle length relation to its diameter, that is the doubled radius. Means, for finding of length of a circle you should increase the value of radius received on the previous step by the doubled Pi's number. In a general view this formula can be written down so: l=4*π*S / (a+b+c).
3. If the area of a triangle is unknown, but the size of one of its corners (α) and lengths of all parties is given (a, b and c), the radius of an inscribed circle (r) it is possible to express through a tangent of angle α. For this purpose at first put lengths of all parties and halve result, then take away length of that party (a) which lies opposite to a corner of the known size from the received value. The received number should be increased by a tangent of a half of the known size of a corner: r=((a+b+c)/2-a) *tg(α/2). If to replace with this formula in the second step expression from the first step, then the formula of length of a circle will take such form: l=2*π * (a+b+c) / 2-a) *tg(α/2).
4. It is possible to manage and only lengths of the parties of a triangle (a, b and c). But in this case for simplification of a formula it is better to enter an additional variable - poluperimetr a triangle: p=(a+b+c)/2. With its help the radius of an inscribed circle can be expressed as a square root from the work of a difference of a poluperimetr, private from division, and length of each of the parties on poluperimetr: r= √ (p-a) * (p-b) * (p-c)/p). And the formula of length of an inscribed circle in this case will take such form: l=2*π * √ (p-a) * (p-b) * (p-c) / p).