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# Relation of Polygon Sides to Angles and Diagonals

## Use the formula (x - 2)180 to find the sum of the interior angles of any polygon.

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Relation of Polygon Sides to Angles and Diagonals

Bradley is helping design an amusement park. He sketches his designs by using regular polygons. He also uses regular polygons to sketch the tables that he wants to place in the picnic area. He sketches a table that has 6 sides. What is the sum of the interior angles?

In this concept, you will learn how to relate the sides of a polygon to angle measures.

### Relating the Sides of a Polygon to Angles and Diagonals

Polygons can be divided into triangles using diagonals. This becomes very helpful when you try to figure out the sum of the interior angles of a polygon other than a triangle or a quadrilateral.

The sum of the interior angles of a quadrilateral is 360\begin{align*}360^\circ\end{align*}.  You can divide a quadrilateral into two triangles using diagonals. Each triangle is 180\begin{align*}180^\circ\end{align*}, so the sum of the interior angles of a quadrilateral is 360\begin{align*}360^\circ\end{align*}.

Here is one diagonal in the quadrilateral.

A diagonal is a line segment in a polygon that joins two nonconsecutive vertices. Divide up the polygon using diagonals and figure out the sum of the interior angles.

Here is a hexagon that has been divided into triangles by the diagonals. You can see here that there are four triangles formed. If the sum of the interior angles of each triangle is equal to 180\begin{align*}180^\circ\end{align*}, and you have four triangles, then the sum of the interior angles of a hexagon is:

4(180)=720\begin{align*}4(180) = 720^\circ\end{align*}

You can follow this same procedure with any other polygon.

If you do not have a picture of the polygon, you can use a formula that involves the number of sides in a polygon.

To better understand how this works, let’s look at a table that shows the number of triangles related to the number of sides in a polygon.

The biggest pattern to notice is that the number of triangles is 2 less than the number of sides. If you know that the sum of the interior angles of one triangle is equal to 180 degrees and if you know that there are three triangles in a polygon, then you can multiply the number of triangles by 180 and that will give you the sum of the interior angles.

Here is the formula.

x=\begin{align*}x =\end{align*} number of sides

(x2)180=\begin{align*}(x - 2)180 =\end{align*} sum of the interior angles

You can take the number of sides and use that as x\begin{align*}x\end{align*}.

Then solve for the sum of the interior angles.

Let’s look at an example.

What is the sum of the interior angles of a decagon?

A decagon has ten sides. That is your x\begin{align*}x\end{align*} measurement. Now let’s use the formula.

(x2)1808(180)=(102)180=1440\begin{align*}(x - 2)180 & = (10 - 2)180 \\ 8(180) & = 1440^\circ\end{align*}

The answer is that there are 1440\begin{align*}1440^\circ\end{align*} in a decagon.

### Examples

#### Example 1

Earlier, you were given a problem about Bradley and his picnic table designs.

He sketches a table that has 6 sides. What is the sum of the interior angles?

First, write the number of sides that are in the polygon.

6

Next, plug the number of sides in to the formula.

(62)(180)\begin{align*}(6-2)(180)\end{align*}(52)(180)\begin{align*}(5-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

4(180)=720\begin{align*}4(180) = 720\end{align*}

The answer is 720o\begin{align*}720^o\end{align*}. The sum of the interior angles is 720o\begin{align*}720^o\end{align*}.

#### Example 2

Calculate the sum of the interior angles of a regular nonagon.

First, write the number of sides that are in a nonagon.

9

Next, plug the number of sides in to the formula.

(92)180\begin{align*}(9 - 2)180\end{align*}

Then, solve for the sum of the interior angles.

(7)(180)=1260\begin{align*}(7)(180) = 1260\end{align*}

The answer is 1260\begin{align*}1260^\circ\end{align*}.

Calculate the sums for each example.

#### Example 3

Calculate the sum of the interior angles of a pentagon.

First, write the number of sides that are in a pentagon.

5

Next, plug the number of sides in to the formula.

(52)(180)\begin{align*}(5-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

(3)(180)=540\begin{align*}(3)(180) = 540\end{align*}

The answer is 540o\begin{align*}540^o\end{align*}.

#### Example 4

Calculate the sum of the interior angles of a triangle.

First, write the number of sides that are in a triangle.

3

Next, plug the number of sides in to the formula.

(32)(180)\begin{align*}(3-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

1(180)=180\begin{align*}1(180) = 180\end{align*}

The answer is 180o\begin{align*}180^o\end{align*}.

#### Example 5

Calculate the sum of the interior angles of an octagon.

First, write the number of sides that are in an octagon.

8

Next, plug the number of sides in to the formula?'

(82)(180)\begin{align*}(8-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

6(180)=1080\begin{align*}6(180) = 1080\end{align*}

The answer is 1080o\begin{align*}1080^o\end{align*}.

### Review

Look at each image and name the type of polygon pictured.

Name the number of diagonals in each polygon.

Use the formula to name the sum of the interior angles of each polygon.

1. Hexagon
2. Pentagon
3. Decagon
4. Heptagon
5. Octagon
6. Square

To see the Review answers, open this PDF file and look for section 9.14.

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### Vocabulary Language: English

TermDefinition
Congruent Congruent figures are identical in size, shape and measure.
Decagon A decagon is a ten sided polygon.
Diagonal A diagonal is a line segment in a polygon that connects nonconsecutive vertices
Heptagon A heptagon is a seven sided polygon.
Hexagon A hexagon is a six sided polygon.
Irregular Polygon An irregular polygon is a polygon with non-congruent side lengths.
Nonagon A nonagon is a nine sided polygon.
Nonconsecutive Nonconsecutive means not next to each other or not in order.
Octagon An octagon is an eight sided polygon.
Pentagon A pentagon is a five sided polygon.
Polygon A polygon is a simple closed figure with at least three straight sides.
Regular Polygon A regular polygon is a polygon with all sides the same length and all angles the same measure.