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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

Have you ever tried to name a figure based on angle measures? Take a look at this problem.

Travis is drawing a design for the skatepark. He has labeled all of the angles in his figure.

The sum of the angle measures is \begin{align*}900^\circ\end{align*}.

What figure is Travis drawing? Do you know how to figure this out?

This Concept will teach you how to relate the sides of a polygons to angle measures. By the end of the Concept, you will know how to help Travis figure this out.

### Guidance

We can divide polygons into triangles using diagonals. This becomes very helpful when we try to figure out the sum of the interior angles of a polygon other than a triangle or a quadrilateral.

Look at the second piece of information in this box. The sum of the interior angles of a quadrilateral is \begin{align*}360^\circ\end{align*}. Why is this important? You can divide a quadrilateral into two triangles using diagonals. Each triangle is \begin{align*}180^\circ\end{align*}, so the sum of the interior angles of a quadrilateral is \begin{align*}360^\circ\end{align*}.

Here is one diagonal in the quadrilateral. We can only draw one because otherwise the lines would cross.

A diagonal is a line segment in a polygon that joins two nonconsecutive vertices.

A consecutive vertex is one that is next to another one, so a nonconsecutive vertex is a vertex that is not next to another one.

How do we use this with other polygons?

We can divide up other polygons 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 sum of the interior angles of each triangle is equal to \begin{align*}180^\circ\end{align*}, and we have four triangles, then the sum of the interior angles of a hexagon is:

We can follow this same procedure with any other polygon.

What if we don’t have the picture of the polygon? Is there another way to figure out the number of triangles without drawing in all of the diagonals? The next section will show you how using a formula with the number of sides in a polygon can help you in figuring out the sum of the interior angles.

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

Do you see any patterns?

The biggest pattern to notice is that the number of triangles is 2 less than the number of sides. Why is this important? Well, 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.

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

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

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

Then solve for the sum of the interior angles.

Let’s try this out.

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

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

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

Try a few of these on your own.

#### Example A

The sum of the interior angles of a pentagon

Solution:

#### Example B

The sum of the interior angles of a triangle

Solution:

#### Example C

The sum of the interior angles of an octagon

Solution:

Here is the original problem once again.

Travis is drawing a design for the skatepark. He has labeled all of the angles in his figure.

The sum of the angle measures is \begin{align*}900^\circ\end{align*}.

What figure is Travis drawing? Do you know how to figure this out?

We can figure this out by using the formula for angle measures and sides of a polygon.

\begin{align*}(x - 2)180 = 900^\circ\end{align*}

Now we can solve this just as we would an equation. Begin by dividing both sides by 180 degrees.

\begin{align*}x - 2 = 900 \div 180\end{align*}

\begin{align*}x - 2 = 5\end{align*}

Next add 2 to both sides of the equation.

\begin{align*}x - 2 + 2 = 5 + 2\end{align*}

\begin{align*}x = 7\end{align*}

Travis' figure is a heptagon with seven sides.

### Guided Practice

Here is one for you to try on your own.

What is the sum of the interior angles of a regular nonagon?

To figure this out, we can use the formula presented in the Concept.

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

In this formula, the value of \begin{align*}x\end{align*} is the number of sides of the polygon.

In this case, a nonagon has 9 sides.

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

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

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

### Explore More

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

1.

2.

3.

4.

5.

6.

Directions: Name the number of diagonals in each polygon.

7.

8.

9.

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

10. Hexagon

11. Pentagon

12. Decagon

13. Heptagon

14. Octagon

15. Square

### Vocabulary Language: English

Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Decagon

Decagon

A decagon is a ten sided polygon.
Diagonal

Diagonal

A diagonal is a line segment in a polygon that connects nonconsecutive vertices
Heptagon

Heptagon

A heptagon is a seven sided polygon.
Hexagon

Hexagon

A hexagon is a six sided polygon.
Irregular Polygon

Irregular Polygon

An irregular polygon is a polygon with non-congruent side lengths.
Nonagon

Nonagon

A nonagon is a nine sided polygon.
Nonconsecutive

Nonconsecutive

Nonconsecutive means not next to each other or not in order.
Octagon

Octagon

An octagon is an eight sided polygon.
Pentagon

Pentagon

A pentagon is a five sided polygon.
Polygon

Polygon

A polygon is a simple closed figure with at least three straight sides.
Regular Polygon

Regular Polygon

A regular polygon is a polygon with all sides the same length and all angles the same measure.