How many diagonals are present in a dodecagon?

To determine the number of diagonals in a dodecagon, which has 12 sides, we can use the formula for calculating the number of diagonals in a polygon:

[

\text{Number of diagonals} = \frac{n(n-3)}{2}

]

Where ( n ) is the number of sides of the polygon.

For a dodecagon:

1. Substitute ( n = 12 ) into the formula:

[

\text{Number of diagonals} = \frac{12(12-3)}{2}

]

2. This simplifies to:

[

\text{Number of diagonals} = \frac{12 \times 9}{2} = \frac{108}{2} = 54

]

Thus, a dodecagon has 54 diagonals. This is calculated by considering that each vertex connects to ( n-3 ) other vertices (excluding itself and the two adjacent vertices), and we account for the fact that each diagonal is counted twice when considering both endpoints, hence the division by 2 in the formula. This systematic approach validates that the total number of diagonals in a