A decagon has how many diagonals?
The correct answer and explanation is :
A decagon has 35 diagonals.
To understand how to calculate this, we first need to understand the formula used to determine the number of diagonals in any polygon. The formula for the number of diagonals ( D ) in a polygon with ( n ) sides (or vertices) is:
[
D = \frac{n(n – 3)}{2}
]
Where:
- ( n ) is the number of sides (or vertices) of the polygon.
- ( D ) is the number of diagonals.
Applying the Formula to a Decagon
A decagon is a polygon with 10 sides. Therefore, ( n = 10 ). Plugging this value into the formula:
[
D = \frac{10(10 – 3)}{2}
]
[
D = \frac{10(7)}{2}
]
[
D = \frac{70}{2}
]
[
D = 35
]
So, a decagon has 35 diagonals.
Why Does This Formula Work?
This formula works because, for any polygon with ( n ) sides, each vertex can connect to ( n – 3 ) other vertices to form diagonals. The reason for ( n – 3 ) is that a vertex cannot connect to itself (that would be a side, not a diagonal), and it also cannot connect to its two adjacent vertices (since those are the sides of the polygon, not diagonals). Hence, each vertex is connected to ( n – 3 ) other vertices to form a diagonal.
However, this method counts each diagonal twice (once from each end), so we divide the result by 2 to avoid double-counting.
Conclusion
For a decagon, the number of diagonals is calculated to be 35. This formula can be applied to any polygon with ( n ) sides to find the number of diagonals efficiently.