Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Hexagon

Step-by-Step Solution

Answer

The hexagon has 9 diagonals.

Step 1: Identify the number of sides in the polygon

We have a hexagon, which has 6 sides. So, n = 6.

Step 2: Substitute n in the formula

We substitute n = 6 into the formula to solve. The formula for finding the number of diagonals in a polygon is \[n*(n-3)/2\]. When we substitute n=6 into the formula, we get \[6*(6-3)/2\].

Step 3: Simplification

Simplify the equation. \(6*(6-3)/2 = 6*3/2 = 9\).

Key Concepts

Polygon Diagonals Formula

Understanding the formula for calculating the number of diagonals in a polygon is a key part of geometry. In essence, a polygon is a closed figure with at least three straight sides and angles. The formula \[n(n - 3) / 2\] gives us the total number of diagonals, where 'n' represents the number of sides in the polygon. This formula can be derived by considering that each vertex can connect to all other vertices except itself and the two adjacent ones, resulting in \[n-3\] possibilities for diagonals. Multiplying this by the total number of vertices gives us the count of diagonals for the entire polygon; however, because each diagonal has been counted twice, we must divide the product by 2 for the correct total.

Geometry

Geometry is the branch of mathematics concerned with questions of shape, size, relative position of figures, and properties of space. It is one of the oldest sciences, dating back to ancient Greek mathematicians like Euclid.
Diagonals are one of many geometric concepts and are particularly interesting as they have properties that can affect both the aesthetic and functional aspects of the polygon. In learning about the geometrical concept of diagonals, we enhance our spatial understanding and problem-solving skills. Diagonals can divide a polygon into triangles, enabling us to calculate areas and other properties more easily, which often aids in the practical application of mathematics in fields such as engineering and architecture.

Hexagon Properties

A hexagon is a six-sided polygon, a very common geometric figure that features several intriguing properties. In a regular hexagon, all angles and sides are equal, making the shape equiangular and equilateral. The internal angles of a regular hexagon are each 120 degrees, summing up to \[720\] degrees in total.

Many students encounter hexagons when learning about tessellation because regular hexagons can perfectly fit together without gaps, a property seen in nature in honeycombs.
Applying the polygon diagonals formula to a hexagon, we find that it has \[6(6 - 3) / 2 = 9\] diagonals, facilitating connections between non-adjacent vertices and offering insights into its internal structure. For complex polygons, knowing the number of diagonals can significantly impact the understanding of the shape's geometry.