Octagon Diagonals Intersection Points Calculation

To find the maximum number of intersection points formed by the diagonals of a convex octagon, we need to determine how many unique points are created by the crossing diagonals inside the polygon.

Understanding Intersection Points

• An intersection point inside a convex polygon is formed when two diagonals cross each other.
• Crucially, each internal intersection point is uniquely determined by selecting four vertices of the polygon. The diagonals connecting pairs of these vertices will intersect at that point.
• Therefore, the problem reduces to finding the number of ways to choose 4 vertices from the total number of vertices of the polygon.

Applying Combinations Formula

The number of ways to choose $ k $ items from a set of $ n $ items (without regard to the order of selection) is given by the combination formula:

$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $

• For a convex octagon, the number of vertices $ n = 8 $.
• We need to choose $ k = 4 $ vertices to form each unique intersection point.

Calculating Intersection Points for Octagon

Using the combination formula with $ n = 8 $ and $ k = 4 $:

$ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} $

Let's calculate the value:

$ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} $

$ \binom{8}{4} = \frac{1680}{24} $

$ \binom{8}{4} = 70 $

Thus, the maximum number of intersection points formed by the diagonals of a convex octagon is 70.