Problem 75 - Precalculus with Limits

Question

In Exercises 73 - 76, find the number of diagonals of the polygon. (A line segment connecting any two non adjacent vertices is called a diagonal of the polygon.) Octagon

Step-by-Step Solution

Answer

The octagon has 20 diagonals.

Step 1: Understand the Problem

We are asked to find the number of diagonals of an octagon. A diagonal of a polygon in a plane is a line segment that joins non-adjacent vertices.

Step 2: Recall the appropriate formula

The formula to calculate the number of diagonals in a polygon is given by \(n(n-3) / 2\), where 'n' represents the number of sides of the polygon.

Step 3: Insert n into the Formula

For an octagon, 'n' equals 8. Therefore, the formula becomes \(8(8-3) / 2\).

Step 4: Simplify the expression

Upon simplifying the expression, we get \(8 * 5 / 2 = 20\).

Key Concepts

Understanding the Octagon

An octagon is a type of polygon that has eight sides. Each side meets another to form the vertices, which are the corner points of the polygon. Because it has eight sides, an octagon naturally forms eight vertices.

It's helpful to visualize an octagon as a stop sign, which typically has the familiar shape. Recognizing an octagon involves counting its sides or noting that it forms eight angles. Understanding these characteristics helps when identifying or solving problems related to octagons, such as finding its diagonals.

Applying the Diagonal Formula

To find the number of diagonals in a polygon, we use the diagonal formula:

• The formula is: \( \frac{n(n-3)}{2} \)
• Here, \( n \) represents the number of sides in the polygon.

For an octagon, which has 8 sides, plug in the value of \( n \) into the formula: \( \frac{8(8-3)}{2} \).

Follow these steps to calculate:

• Subtract 3 from 8 to get 5.
• Multiply 8 by 5 to get 40.
• Divide 40 by 2, resulting in 20.

This means an octagon has 20 diagonals. The calculation is straightforward once you understand the formula, which is derived by considering that each vertex can connect to \( n-3 \) other vertices except itself and its adjacent neighbors.

Significance of Non-Adjacent Vertices

The term "non-adjacent vertices" is crucial when understanding diagonals of polygons. In a polygon, a vertex is a point where two sides meet. Non-adjacent vertices refer to any other vertices that are not directly connected by a side.

When you draw a diagonal, you are connecting two of these non-adjacent vertices. For example, in an octagon, if you start at one vertex and skip the next immediate vertex to draw a line, you've drawn a diagonal.

Understanding what non-adjacent means helps avoid errors in counting diagonals. In summary, connecting all possible pairs of non-adjacent vertices gives the total number of diagonals in any polygon, using the diagonal formula.