Perimeter of a polygon call the closed broken line made of all its parties. Finding of length of this parameter comes down to summation of lengths of the parties. If all pieces forming perimeter of such two-dimensional geometrical figure have the identical sizes, the polygon is called regular. In this case calculation of perimeter considerably becomes simpler.

## Instruction

1. In the simplest case when length of the party (a) of a regular polygon and number of tops (n) in it are known, for calculation of length of perimeter (P) just multiply these two sizes: Р = a*n. For example, length of perimeter of the correct hexagon with the party in 15 cm has to be equal to 15*6=90 cm.

2. It is possible to calculate perimeter of such polygon on the known radius (R) of the circle described about it too. For this purpose it is necessary to express at first length of the party with use of radius and quantity of tops (n), and then to increase the received size by number of the parties. To calculate length of the party increase radius by a sine of the number of Pi divided into quantity of tops, and double result: R*sin(π/n)*2. If it is more convenient to you to calculate trigonometrical function in degrees, replace Pi's number with 180 °: R*sin (180 ° / n) *2. Calculate perimeter multiplication of the received size by number of tops: Р = R*sin(π/n) *2*n = R*sin (180 ° / n) *2*n. For example, if the hexagon is entered in a circle with a radius of 50 cm, its perimeter will have length 50*sin (180 °/6)*2*6 = 50*0.5*12 = 300 cm.

3. A similar way it is possible to consider perimeter, without knowing length of the party of a regular polygon if it is described about a circle with the known radius (r). In this case the formula for calculation of the size of the party of a figure will differ from only in the previous used trigonometrical function. Replace in a formula a sine with a tangent to receive such expression: r*tg(π/n)*2. Or for calculations in degrees: r*tg (180 ° / n) *2. For calculation of perimeter increase the received size in the number of times equal to quantity of tops of a polygon: Р = r*tg(π/n) *2*n = r*tg (180 ° / n) *2*n. For example, the perimeter of the octagon described near a circle with a radius of 40 cm will be approximately equal 40*tg (180 °/8)*2*8 ≈ 40*0.414*16 = 264.96 cm.