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How many diagonals are there in a polygon of 12 sides ?
1. 78
2. 66
3. 54
4. 48
5. None of these

1 Answer

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Best answer
Correct Answer - Option 3 : 54

Concept:

The number of ways to select r things out of n things is given by nCr

\(\rm ^nC_r=\frac{n!}{(n-r)!\times(r)!}=\frac{n\times(n-1)\times....(n-r+1)}{r!}\)

 

Calculation:

A polygon of 12 sides has 12 vertices.

By joining any two of the vertices, we obtain either a side or a diagonal of the polygon.

Number of all straight lines obtained by joining 2 vertices at a time = \(\rm ^{12}C_2=\frac{12\times11}{2\times1}\)

= 66

These straight lines include 12 sides of the polygon.

∴ The number of diagonals of the polygon = 66 - 12

= 54

Hence, option (3) is correct. 

 

The number of diagonals in a polygon of n sides is given by, \(\rm ^nC_2-n\) 

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