1 . How many diagonals does each of the following have?

To determine the number of diagonals in a polygon, we can use a straightforward formula. The formula for finding the number of diagonals in a polygon with \( n \) sides is given by:

Diagonal Formula

The formula is:

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

Here, \( n \) represents the number of sides (or vertices) of the polygon. This formula works because each vertex connects to \( n - 3 \) other vertices to form a diagonal (you can't connect to itself or the two adjacent vertices). Since each diagonal is counted twice (once from each end), we divide by 2.

Example Calculations

Let’s apply this formula to a few different polygons:

• Triangle (3 sides):
  Using the formula:
  Diagonals = \( \frac{3(3 - 3)}{2} = \frac{3 \times 0}{2} = 0 \)
  A triangle has no diagonals since all its vertices are connected directly.
• Quadrilateral (4 sides):
  Diagonals = \( \frac{4(4 - 3)}{2} = \frac{4 \times 1}{2} = 2 \)
  A quadrilateral has 2 diagonals.
• Pentagon (5 sides):
  Diagonals = \( \frac{5(5 - 3)}{2} = \frac{5 \times 2}{2} = 5 \)
  A pentagon has 5 diagonals.
• Hexagon (6 sides):
  Diagonals = \( \frac{6(6 - 3)}{2} = \frac{6 \times 3}{2} = 9 \)
  A hexagon has 9 diagonals.
• Heptagon (7 sides):
  Diagonals = \( \frac{7(7 - 3)}{2} = \frac{7 \times 4}{2} = 14 \)
  A heptagon has 14 diagonals.

Understanding the Concept

Visualizing this can help. Imagine a pentagon. Each vertex connects to two adjacent vertices (which are not diagonals) and two non-adjacent vertices (which are diagonals). This is why the formula accounts for the connections that do not form diagonals.

In summary, the number of diagonals increases as the number of sides increases, and you can easily calculate it using the formula provided. If you have a specific polygon in mind, feel free to share, and we can calculate the diagonals together!