Question

Calculate the maximum number of diagonals that can be drawn in an octagon using the suitable formula.

Answer 1

Hint: Here the n-sided polygon given to us is an eight sided polygon, that is an octagon.So an octagon will have 8 equal sides. For a n-sided polygon, the maximum number of diagonals are calculated with the formula of combinations gives as:
${}^n{C_2} - n$, where n are the number of sides or vertices.

Complete step by step answer:
Now, in an octagon the number of vertices and sides are 8.
So every vertex will have 5 diagonals connecting to the opposite vertex.
Now, for a n-sided polygon the formula of the number of diagonals is given by the formula of Combination as:
${}^n{C_2} - n$, where n are the number of sides or vertices. On expanding and simplifying this will become:
$
  {}^n{C_2} - n \\
   = \dfrac{{n(n - 3)}}{2} \\
$
Now for an octagon, the number of sides is 8,which means n is 8. So substituting n = 8,
The maximum number of possible diagonals that can be drawn are:
$
  {}^8{C_2} - 8 \\
   = \dfrac{{8(8 - 3)}}{2} \\
   = \dfrac{{8 \times 5}}{2} \\
   = 20 \\
$
So, the maximum number of diagonals that can be drawn in an octagon are 20.

Hence, a maximum of 20 diagonals can be drawn in an octagon.

Note: A regular n-sided polygon is a geometrical shape drawn with n number of equal sides. There are special names for these polygons. For example, a 3-sided polygon is called a triangle; a 5-sided polygon is called a pentagon. Similarly an 8-sided polygon is called an octagon. The number of diagonals is basically, the lines drawn from one vertex to the rest of the opposite vertices.