Number of diagonals in a polygon

A diagonal of a polygon is a line segment that connects two vertices that are not next to each other.

Use the canvas below to find the total number of line segments, and the number of diagonals for each polygon.

How many segments will there be in a 20-sided polygon (icosagon)? How many of them will be the diagonals of the icosagon? Try to write a general term for the total number of line segments and the number of diagonals of any polygon.

Thinking about these questions can help you to find a general rule about the number of diagonals.

• What is your strategy to count the total number of segments for each polygon? What did you realize?
• Is there a number pattern for the total number of segments? How many of them are the sides and how many are the diagonals?
• What is the number of segments you draw from each vertex? How is it related to the total number?
• What is the number of diagonals you can draw per vertex?
• How can we avoid double counting when finding the total number of diagonals?

Solution

There are several ways of writing a general rule for an n-sided polygon.

You may recognize the pattern for the total number of segments:

3-6-10-15-21-28-36-45 ..

It is the sequence of triangular numbers starting from 3.

One way to express the total number of segments for an n-gon is n(n−1)/2n(n-1)/2n(n−1)/2 . Since there are n sides, the remaining n(n−1)/2−nn(n-1)/2 -nn(n−1)/2−n of them are the diagonals of a convex polygon.

Another way to express the general rule for the total number of diagonals is to think about the number of diagonals that can be drawn from each vertex in a polygon.

To find out how many diagonals a polygon has, first count the number of sides, or straight lines, that make up the polygon. Then, subtract 3 from the number of sides. Next, multiply that number by the number of sides. Finally, divide the answer by 2, and you’ll have the number of diagonals within the polygon. For example, if a polygon has 6 sides, you’d find it has 9 diagonals. For an alternate way to determine the number of diagonals in a polygon, read on!

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The diagonal of a polygon is a line segment that connects any two non-adjacent vertices. The number of diagonals and their attributes vary depending on the type of polygon and the number of sides. Let's review what a polygon is and what a diagonal is before learning the diagonal of a polygon formula. 

A closed shape made up of three or more line segments is called a polygon. A line segment generated by joining any two non-adjacent vertices forms the diagonal of a polygon. Let's look at the formula for a polygon's diagonal, as well as some examples of solved problems. You can quickly count all of the possible diagonals of a basic polygon with a few sides. Counting polygons can be difficult when they become more intricate.

Fortunately, there is a straightforward formula for calculating the number of diagonals in a polygon. Because any vertex (corner) is connected to two other vertices by sides, those connections cannot be considered diagonals. That vertex, too, is unable to make a connection with itself.  As a result, ‘n’ we'll immediately lower the number of viable diagonals by three. 

That vertex, too, is unable to make a connection with itself. For example, our door only has two diagonals if you don't include moving from the top hinge to the bottom opposite and back. Any solution will have to be divided by two.

Diagonal- Polygons Diagonals

A diagonal is a segment of a polygon that connects two non-consecutive vertices. In a polygon, the number of diagonals that can be drawn from any vertex is three less than the number of sides. Multiply the number into totaling of diagonals per vertex (n - 3) by the number of vertices, n, then divide by 2 to get the total number of diagonals in a polygon (otherwise each diagonal is counted twice).

Number of Diagonals = \[\frac{n(n-3)}{2}\]

Simply subtract the total sides from the diagonals given by each vertex to another vertex to arrive at this formula. To put it another way, an n-sided polygon has n-vertices that can be connected in nC2 ways. 

The formula obtained by subtracting n using nC2 methods is \[\frac{n(n-3)}{2}\]. 

The total sides of a hexagon, for example, are six. As a result, the total diagonals are 6(6-3)/2 = 9

Let’s know what a diagonal is. A diagonal of a polygon can be defined as a line segment joining two vertices. From any given vertex, there are no diagonals to the vertex on either side of it, since that would lay on top of the side. Also,  remember that there is obviously no diagonal from a vertex back to itself. This means there are three less diagonals than the number of vertices. (We do not count diagonals to itself and one either side). This is a diagonal definition.

Here, we are going to discuss the number of diagonals in a polygon, diagonal definition.

Formula for the Number of Diagonals

As described above, the number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).

There are a total number of N vertices, which gives us n(n-3) diagonals.

But each diagonal of the polygon has two ends, so this would count each one twice. So as a final step we need to divide by 2, for the final formula:

Number of distinct diagonals = \[\frac{n(n-3)}{2}\]

where,

n  is the number of sides (or vertices).

Diagonals of Polygon

The diagonals of a polygon is a segment line in which the ends are non-adjacent vertices of a polygon.

How many diagonals does n-polygon have? Let’s see the diagonals of a polygon and the no. of diagonals in a polygon.

For n = 3 we have a triangle. We can clearly see the triangle has no diagonals because each vertex has only adjacent vertices. Therefore, the number of diagonals in a polygon triangle is 0.

For n = 4 we have quadrilateral.  It has 2 diagonals. Therefore, the number of diagonals in a polygon quadrilateral is 2.

For n = 5, we have a pentagon with 5 diagonals. Therefore, the number of diagonals in a polygon pentagon is 5.

For n = 6, n-polygon is called hexagon and it has 9 diagonals. Therefore, the number of diagonals in a polygon hexagon is 9.

Since n was a lower number we could easily draw the diagonals of n-polygons and then count the number of diagonals in a polygon.

Diagonals in Real Life

Diagonals in rectangles, as well as diagonals in squares, add toughness to construction, whether for a house wall, bridge, or tall building. You may have seen diagonal wires used to keep the bridges steady. When houses are being built, look for the diagonal braces that tend to hold the walls straight and true.

Bookshelves and scaffolding are braced with diagonals. For a catcher in softball or for a catcher in baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.

The phone screen or computer screen you are viewing this lesson on is measured along its diagonal. A 21" screen never tells you the width and height of the screen; it is 21" from one corner to an opposite corner.

Diagonal Formulas

1) Diagonal of a Rectangle Formula:

Diagonal of a Rectangle = \[\sqrt{l^{2} + b^{2}}\]

For rectangles, l is the length of the rectangle and b is the breadth of the rectangle.

2) Diagonal of a Square Formula:

Now let's look at a few different diagonal formulas to find the length of a diagonal of a square.

Diagonal of a Square = \[a\sqrt{2}\]

For squares, a is the side of a square

3) Diagonal of a Cube Formula:

For any given cube, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem/distance formula:

Diagonal of a cube = \[\sqrt{s^{2} + s^{2} + s^{2}}\]

For cubes, s is the side of a cube

Important Note

The above-given formula gives us the number of distinct diagonals - that is, the number of actual line segments. At times it is easy to miscount the diagonals of a polygon when doing it by eye.

If you glance quickly at the pentagon given below, you may be tempted to say that the number of diagonals is 10. After all, there are 2 at each vertex and 5 vertices. Few people watch them making 3 triangles, for 6 diagonals. But there are only 5 diagonals. You need to count them carefully.

Question 1) Find the total number of diagonals contained in an 11-sided regular polygon.

Solution) In an 11-sided polygon, total vertices are 11. Now, the 11 vertices can be joined with each other by C211 ways i.e. 55 ways.

Now, there are a total of 55 diagonals possible for an 11-sided polygon which includes its sides also. So, subtracting the sides will give the total number of diagonals contained by the polygon.

How many diagonals does a 5 sided polygon have?

What is diagonal of a polygon?

What is the formula for finding diagonal?

What is the number of diagonals in a polygon of 20 sides?