In octagon perimeter, as well as any other flat geometrical figure, call the sum of lengths of its parties. It is necessary to solve a problem of determination of this parameter of a polygon sometimes only with use of mathematical formulas, and sometimes - to measure them by any make-shifts. Anyway ways of the solution of a task exists a little and each of them will be optimum in relation to a certain set of initial conditions.
Instruction
1. If it is necessary to calculate perimeter (P) of an octagon in the theory, and in initial conditions lengths of all parties of this figure are given (a, b, c, d, e, f, g, h), put these sizes: P = a+b+c+d+e+f+g+h. It is necessary to know lengths of all parties only in case of the wrong polygon and if from statements of the problem it is known that the figure is correct, then there will be enough length of one party - just increase it by eight times: P = 8*a.
2. If in basic data nothing is told about length of the party of the correct octagon, but the radius of this figure of a circle (R) described near is specified, then before application of a formula from the previous step it is necessary to calculate a missing variable. Each of the parties in such octagon can be considered the basis of an isosceles triangle which sides are radiuses of a circumscribed circle. As all such identical triangles there will be eight, corner size between radiuses of each of them will be the one eighth part from a whole revolution: 360 °/8 = 45 °. Knowing lengths of two parties of a triangle and size of a corner between them, determine basis size - increase a cosine of a half of a corner by the doubled side length: 2*R*cos (22.5 °) ≈ 2*R * 0.924 ≈ R * 1.848. Substitute the received value in a formula from the first step: P ≈ 8*R*1.848 ≈ R*14.782.
3. If in statements of the problem only a radius (r) of the circle entered in the correct octagon is given, then it is necessary to make the calculations similar to described above. In this case radius can be presented as one of legs of a rectangular triangle which other leg will be a half from the party of an octagon necessary to you. The acute angle adjoining radius will be twice smaller calculated in the previous step: 360 °/16 = 22.5 °. Calculate length of the necessary leg multiplication of a tangent of this corner by other leg (radius), and for determination of size of the party of an octagon double the received value: 2*r*tg (22.5 °) ≈ 2*r*0.414 ≈ r*0.828. Substitute this expression in a formula from the first step: P ≈ 8*r*0.828 ≈ r*6.627.
4. If it is required to calculate radius by method of practical measurements, then, depending on the figure size, use, for example, a ruler, the curvimeter ("a roller range finder") or a pedometer. Substitute the received values of lengths of the parties in one of two formulas given in one of steps.