A regular polygon with 'n' vertices has one of
its side length equal to 's' units. What will be
the average length (in units) of all the sides of
the regular polygon?
Answers
Step-by-step explanation:
3.1 Regular Polygons
The complete set of regular polygons is comprised of all structures having n equal sides and n equal angles, where n is any integer from three to infinity. At n = 3, the structure is an equilateral triangle, n = 4 describes a square, and so forth, to n = ∞, which represents a circle.
The properties of each of these structures can be established by analysis of the set of identical triangles formed by radiating lines from the center of the polygon to each of the corners.
The geometric surface area and open frontal area for the regular polygons are expressed by the following equations:
(5)GSA=2⋅n⋅N⋅h⋅ctn[90−(180/n)]
(6)OFA=n⋅N⋅h2⋅ctn[90−(180/n)]
where h is the distance from the center of the polygon to the center of an inside open side and n is the number of sides. For this class of structures, the hydraulic diameter is the same as the diameter of the largest cylinder that can be inserted into the channel.
Although these relationships can be written for all members of this set, only three regular polygons - the triangle, square, and hexagon - can be assembled at arbitrary values of open frontal area. Hence, although it is interesting to observe the relationships for the other structures of this set, only these three are of practical importance for the automotive catalyst support application.
In all of the figures relating to the regular polygons, the values will be given as discrete points without connecting lines because the regular polygons are represented by a series of integers and not, as with the other sets of shapes discussed, a continuous set of real numbers.
The remaining information necessary to derive the heat&mass transfer and pressure drop relationships for the regular polygons are the Nusselt Number and Friction Factor (9) for each of these shapes. A set of data for these two shape related quantities is given in Figure 1, where the Friction Factor and average Nusselt Number are plotted against a shape value [1/(1+1/n)], where n is the number of sides. In Figure 1 the values of n are 3 through 10, 20, and infinity.
Given: A regular polygon with 'n' vertices has one of its side length equal to 's' units.
To find: Average length of all the sides
Explanation: Number of sides= n
Length of each side = s
In a regular polygon, all angles as well as sides are equal.
Total length of all the sides will be equal to the sum of all the sides.
Sum of all sides is
= product of the number of sides and length of each side.
= n × s
Average length is equal to the total length of the sides divided by the total number of sides.
Average length
= Total length/ Number of sides
= n× s / n
= s
Therefore, the average length of all the sides is s.