The number of diagonals that can be drawn by joining the vertices of an octagon, is

A) 28

B) 48

C) 20

D) None of these

To determine the number of diagonals in an octagon, we can use a specific formula that applies to any polygon. The formula for calculating the number of diagonals in a polygon with \( n \) sides is given by:

Diagonal Formula

The formula is:

Diagonals = \frac{n(n - 3)}{2}

In this case, since we are dealing with an octagon, \( n \) is equal to 8. Let's substitute 8 into the formula.

Calculating the Number of Diagonals

• First, substitute \( n \) with 8:
• Diagonals = \(\frac{8(8 - 3)}{2}\)
• Now simplify the expression:
• Diagonals = \(\frac{8 \times 5}{2}\)
• Diagonals = \(\frac{40}{2}\)
• Diagonals = 20

Thus, the number of diagonals that can be drawn by joining the vertices of an octagon is 20. Therefore, the correct answer is C) 20.

Understanding the Concept Further

To grasp why this formula works, consider that each vertex of the octagon can connect to other vertices to form diagonals. However, a vertex cannot connect to itself or its two adjacent vertices (as those connections would form sides of the octagon, not diagonals). This restriction is why we subtract 3 from \( n \) in the formula.

Visualizing with an Example

Imagine labeling the vertices of the octagon as A, B, C, D, E, F, G, and H. From vertex A, you can draw diagonals to vertices C, D, E, F, G, and H, but not to B or A itself. This gives you 5 diagonals from vertex A alone. If you repeat this process for each vertex, you would initially count each diagonal twice (once from each end), which is why we divide by 2 in the formula.

In summary, the number of diagonals in an octagon is 20, and this can be calculated efficiently using the formula \( \frac{n(n - 3)}{2} \). This approach can be applied to any polygon, making it a versatile tool in geometry.