Question

How many diagonals can a regular hexagon have? (1)

a) 18 b) 12

c) 9 d) 6​

Answer 1

Answer:

ans is option (c) 9

Explanation:

Because in a hexagon , the total sides are 6.

so, the total diagonals will be 6 (6-3)/2 = 9.

Answer 2

Suppose we've n points in a plane, which are arranged in such a way to make an n sided regular polygon including its diagonals.

Making a diagonal or a side of the polygon means selecting any two among the $n$ points and joining them with a straight line.

No. of ways of selecting two among the n points $=\,^n\!C_2$

No. of ways of joining the selected two points $=\,^2\!C_2=1.$

Hence no. of ways of making a side or a diagonal of the regular polygon $=\,^n\!C_2\times1=\,^n\!C_2.$

We know the regular polygon has n sides, So the no. of diagonals will be,

$\longrightarrow d=\,^n\!C_2-n$

$\longrightarrow d=\dfrac{n(n-1)}{2}-n$

$\longrightarrow d=\dfrac{n^2-n-2n}{2}$

$\longrightarrow d=\dfrac{n^2-3n}{2}$

$\longrightarrow\boxed{d=\dfrac{n(n-3)}{2}}$

In the question, it's given a regular hexagon, so $n=6.$

Hence no. of diagonals in a regular hexagon is,

$\longrightarrow d=\dfrac{6(6-3)}{2}$

$\longrightarrow d=\dfrac{6\times3}{2}$

$\longrightarrow\underline{\underline{d=9}}$

Hence (c) is the answer.