Problem 74 - 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.) Hexagon

Step-by-Step Solution

Answer

The number of diagonals in a hexagon is 9.

Step 1: Understanding the Problem

You are given a hexagon, which is a polygon with 6 sides. Non-adjacent vertices are vertices that are not directly linked by a side. A diagonal is a line segment that connects any two non-adjacent vertices of a polygon. The problem asks for the total number of these diagonals.

Step 2: Applying the Diagonal Formula

Use the formula for finding number of diagonals in a polygon, which is \(n(n-3)/2\). Here, n is the number of sides in the polygon. For a hexagon, n is 6.

Step 3: Calculate the Number of Diagonals

Plug the value into the formula: \(6(6-3)/2\), which simplifies to \(6(3)/2=9\). Hence, a hexagon has 9 diagonals.

Key Concepts

Hexagon

A hexagon is a special type of polygon. It has exactly six sides. Each point where two sides meet is called a vertex. Since hexagons have six sides, they also have six vertices. When you think of a standard hexagon, like a regular one found in beehives, each side is of equal length and angles are all the same. However, hexagons can also be irregular, with different angles and side lengths. What defines a shape as a hexagon is simply having six sides and six vertices, whether they are equal or not.

Polygon

A polygon is a 2-dimensional shape that is made up of straight line segments. These segments are called sides, and they must connect at their endpoints to form a closed shape. The simplest polygons are triangles, which have three sides. As you add more sides, the polygon changes, becoming a quadrilateral, pentagon, hexagon, and so on. Each of these has a specific number of sides and vertices. Importantly, a polygon must not have any sides that cross over — it must be flat. Polygons can be regular, with all sides and angles equal, or irregular, with varying lengths and angles.

Vertices

Vertices are the corner points of a polygon where two sides meet. Every polygon will have a number of vertices equal to its number of sides. So, a triangle has three vertices, a quadrilateral has four, and a hexagon, like the one in our exercise, has six. When calculating diagonals, it's crucial to understand which vertices are adjacent (connected directly by a side) and which are non-adjacent (not directly connected). Diagonals connect non-adjacent vertices.

Diagonal Formula

Diagonals are line segments connecting two non-adjacent vertices of a polygon. The formula to find the number of diagonals in any polygon is \[ \frac{n(n-3)}{2} \]where \(n\) is the number of sides, or vertices, of the polygon. This formula first calculates the total number of potential vertex connections, then subtracts the polygon's sides and divides by two to avoid double-counting. Let's see this in action with a hexagon: \(n\) is 6, so substituting into the formula gives \[ \frac{6(6-3)}{2} = \frac{18}{2} = 9 \]This result shows that a hexagon has 9 diagonals.