7th Grade Math > Properties of Shapes > Principles for Calculating the Number of Diagonals in a Polygon, Practice Worksheets

Diagonals

A diagonal is a line segment connecting two non-adjacent vertices in a polygon.

Understanding the concept of diagonals is crucial because the number of diagonals in a polygon is related to the number of its vertices.

To calculate the number of diagonals, we first need to determine the number of diagonals that can be drawn from a single vertex. Once we know this, we can use the same rule for each vertex in the polygon.

Let's examine this by looking at polygons with different numbers of vertices, such as quadrilaterals, pentagons, and hexagons.

In a quadrilateral, there is 1 diagonal from each vertex. In a pentagon, there are 2 diagonals from each vertex. In a hexagon, there are 3 diagonals from each vertex. As the number of vertices increases by one, the number of diagonals from each vertex also increases by one.

So, how many diagonals can be drawn from a vertex in an n-sided polygon?

Looking at the pattern:

• In a quadrilateral (4 sides), there is 1 diagonal from each vertex.
• In a pentagon (5 sides), there are 2 diagonals from each vertex.
• In a hexagon (6 sides), there are 3 diagonals from each vertex.

Thus, we can generalize that the number of diagonals from one vertex in an n-sided polygon is $n - 3$.

Formula for the Number of Diagonals from One Vertex

$n - 3$

Now, to find the total number of diagonals in a polygon, we need to consider the diagonals from each vertex. Let's calculate this for a pentagon:

In a pentagon, there are $5 - 3 = 2$ diagonals from each vertex, and there are 5 vertices.

$2 \times 5 = 10$

However, this isn't the final answer. Why not?

Because each diagonal is counted twice, once from each end. So, we must divide the total by 2:

$\frac{2 \times 5}{2} = 5$

Thus, a pentagon has a total of 5 diagonals.

For larger polygons like octagons, decagons, or icosagons, we need a formula to make calculations easier.

Formula for the Total Number of Diagonals in an n-sided Polygon

$\frac{n \times (n - 3)}{2}$

Let's solve some examples:

1. For $n = 7$:

$\frac{7 \times (7 - 3)}{2} = 14$

2. For $n = 9$:

$\frac{9 \times (9 - 3)}{2} = 27$

This formula simplifies our calculations significantly. Without it, we would have to draw and count all the diagonals manually, which would be tedious and error-prone.

Understanding the principles behind the formula is crucial for grasping how it works.

Summary

We have explored the formula and principles for calculating the number of diagonals in a polygon.

Below is a sample problem generated by the 'Modoo Math' site, which allows you to create customized practice worksheets.