<p> Enjoy about the facts in a regular polygon With successive diagonals -- Acharya, E.R Associate professor, FOE Abstract</p><p>The lines joining the opposite vertex of the polygonal geometrical figures are called diagonals. The number of diagonals in each geometrical figure is different. The maximum number of such diagonals is known by the name of successive diagonals. Through the using of such diagonals and their relation to each other we get so many facts which lead us to enjoy in a polygon with mathematical sense.</p><p>Key words</p><p>Polygon, successive diagonal, order, peri-cyclic quadrilaterals, compartments, bits.</p><p>General Background</p><p>We have seen and drawn different geometrical figures. Such figures are recognized by different names for example a figure with three sides is a triangle, with four sides is a quadrilateral on the other hand a plane figure with five or more than five sides is known as a polygon. In any regular polygonal geometrical shape the lines joining the opposite vertex are called diagonals.</p><p>Explanation and Derivation</p><p>The total numbers of diagonals in any regular polygonal geometrical figure are different. The maximum number of such diagonals is known by the name of successive diagonals. The following table illustrates the number of such diagonals.</p><p>Polygon Quadrilateral Pentagon Hexagon Heptagon Octagon</p><p>No. of diagonals 2 5 9 14 20 In above table, we see that the number of diagonals are 2, 3, 4, 5, 6,….., respectively. It is note that the diagonal in a triangle is 0. The determination of the number of successive diagonals can be generalized as by the following formula.</p><p>(i) Total No. of successive diagonals (Maximum no. of diagonals)</p><p>N(N-3) = , where sides of complete graph (polygon) N =3, 4, 5, 6, that is N  3. The 2 sequence 狁2,3,4,5,6,..... showing the differences of the number of diagonals shows the order of each vertex of the polygon as a complete graph. For example, the order of each vertex of a triangle is 2. The order of a vertex means the edge ends at that vertex. In this way the order of each vertex of a quadrilateral with successive diagonals is 3. Similarly for pentagon the order of each vertex is 4 and so on. This is illustrated by the following facts. If we consider any polygon as a complete graph then the order of each vertex is obtained from the following formula,</p><p>(ii) Order of vertex of complete polygon </p><p>= Number of sides of regular polygon - one. i.e., O (V) = N-1, where N is the no. of sides of a polygon.</p><p>By drawing all possible diagonals, they would be intersected to each other the there are formed various quadrilaterals inside the polygon and there number can be determined by,</p><p>N(N-1)(N-2)(N-3) (iii) Total Number of peri-cyclic quadrilaterals (Q) = , where N 24 is the number of sides of a polygon. The point of intersection of the successive diagonals is also equal to the no. of pericyclic quadrilateral and is obtained by, N(N-1)(N-2)(N-3) (iv) Total No. of point of intersection of successive diagonals = 24</p><p>D(D+1) = , where N is the number of sides of a polygon, D is the no. of successive 6 diagonals. Since by drawing the successive diagonals we get different compartments having various area and then number of these compartments can be counted by the following formula,</p><p>(D+1)(D+6) 6+7D+D2 (v) Total No. of compartments = = 1+D+Q = , where D is 6 6 the no. of successive diagonals and Q is the peri-cyclic quadrilateral of a polygon.</p><p>Next we have to determine the number of bits into which the diagonals of a polygon divide each other. Since each intersection gives us two bits, and each diagonal gives us one bit. Thus the total number of bits is calculated by,</p><p>D(D+4) (vi)Total no. of bits of the successive diagonals in a polygon = D + 2Q = , 3 where D is the no. of successive diagonals and Q is the no. of peri-cyclic quadrilateral. We can be practicing it by drawing the different polygons.</p><p>Conclusion</p><p>In any complete regular polygon we can find so many facts with the relations among the successive diagonals. The various facts in a regular polygon with these relations help to enjoying us as playing games in polygons. </p><p>References:</p><p>E.R. Acharya, “Curriculum Evaluation and Teaching Method”, Mathematics Education Forum, Council for Mathematics Education, 2004. ------, “Complete Regular Polygonal Graph and Successive Diagonals”, Proceedings of National Conference of Mathematics (NCM 2010), Nepal Mathematical Society, Tribhuvan University, Kathmandu University, Purvanchal University.</p><p>------, “Product of successive half turns and its applications on paper folding”, JMC Research Journal, 2010, Janamaitri Research Committee, Janamaitri Multiple Campus, Kathmandu.</p><p>J. Travers, “Problems Connected with a Regular polygon of N sides”, The Mathematical Gazette, vol. 17, No.226 (1933).</p><p> http:/www.jstor.org/pss/3606507</p>