### Let’s Think About It

Dylan is exploring the geodesic dome. He’s figured out that the average dome has isosceles triangles in it. These triangles, like all triangles, have a sum of 180° for angle measures. Now, Dylan wants to construct a geodesic dome.

As he begins to draw his design, he figures out that he will need a variety of hexagons and pentagons. Dylan is stuck. He can’t figure out how many degrees there will be in each hexagon and how many degrees there will be in each pentagon.

In this concept, you will learn to identify polygons.

### Guidance

A **polygon** is a two-dimensional closed figure that has three or more straight sides. Any figure with straight edges, such as a triangle or rectangle, is a polygon. Figures that have any curved sides or open sides are not classed as polygons.

The following figure is NOT a polygon.

Polygons have special properties that determine their angle and side relationships. For instance, the number of sides a polygon has is related to the number of angles it has, and therefore determines the sum of its angles.

Now that we can distinguish polygons from other figures, let’s take a closer look at them. In general, there are two kinds of polygons: regular polygons and irregular polygons.

**Regular polygon**s have sides and angles that are all congruent. If a regular polygon has five congruent sides then it also has five congruent angles. The number of sides of a regular polygon equals the number of interior angles. As long as the sides are congruent and the angles are congruent, the figure is a regular polygon.

**Irregular polygons** are polygons that do NOT have congruent sides and angles. They are still polygons because they have straight, closed sides. The sides of irregular polygons are different lengths. The following diagram shows a regular and an irregular polygon.

Each polygon shown in the diagram has six sides and six interior angles. Both of the polygons are hexagons. The first hexagon has side sides that are equal in length and six interior angles that are equal in measure. This is a regular hexagon. The second hexagon has six sides that are all different lengths and six interior angles that are different measures. This is an irregular hexagon.

Polygons are named according to the number of straight, closed sides. The number of sides and the number of congruent interior angles are the same number. The number of diagonals of an \begin{align*}n\end{align*}-sided polygon can be found using the formula:

\begin{align*}\# \ \text{of diagonal} = [n (n-3)] \div 2\end{align*}

The sum of the measures of the interior angles of a polygon are different for each polygon. The sum of the interior angles of an

-sided polygon can be found using the formula:\begin{align*}\text{Sum of interior angles} = 180^{\circ} (n-2)\end{align*}

Let’s apply these formulas to the following hexagon to determine the number of diagonals and the sum of its interior angles.

The hexagon is an \begin{align*}n\end{align*}-sided polygon such that the value of ‘\begin{align*}n\end{align*}’ is 6. The diagonals of a polygon are line segments from one corner to another non-adjacent corner.

The number of diagonals can be calculated using the formula:

\begin{align*}\begin{array}{rcl} \# \ \text{of diagonals} &=& [n(n-3)] \div 2 \qquad \qquad \ \text{Where} \ n = 6. \\ \# \ \text{of diagonals} &=& [6 (6-3)] \div 2 \qquad \qquad \ \text{Substitute} \ 6 \ \text{for} \ n \\ \# \ \text{of diagonals} &=& [6(3)] \div 2 \qquad \qquad \qquad \text{Perform the subtraction inside the parenthesis} \\ \# \ \text{of diagonals} &=& 18 \div 2 \qquad \qquad \qquad \quad \ \text{Perform the multiplication inside the square brackets} \\ \# \ \text{of diagonals} &=& 9 \qquad \qquad \qquad \qquad \quad \ \ \text{Perform the division.} \end{array}\end{align*}

The following diagram shows the nine diagonals of the hexagon.

The sum of the interior angles of an

-sided polygon can be calculated using the formula:\begin{align*}\text{Sum of interior angles} = 180^{\circ} (n-2)\end{align*}. Where ‘\begin{align*}n\end{align*}’ is the number of sides of the polygon.

The number of triangles formed by the diagonals of an \begin{align*}n\end{align*}-sided polygon is \begin{align*}(n-2)\end{align*} or two less than the number of sides. Remember, the sum of the interior angles of a triangle is 180°.

\begin{align*}\begin{array}{rcl} \text{Sum of interior angles} &=& 180^{\circ} (n-2) \qquad \ \ \ \ \qquad \text{Where the value of} \ n \ \text{for a hexagon is} \ 6. \\ \text{Sum of interior angles} &=& 180^{\circ} (6-2) \qquad \quad \qquad \ \text{Perform the subtraction inside the parenthesis.} \\ \text{Sum of interior angles} &=& 180^{\circ} (4) \qquad \qquad \qquad \ \ \ \text{Perform the multiplication} \\ \text{Sum of interior angles} &=& 720^{\circ} \end{array}\end{align*}

The simplest way to record the properties of the various regular polygons is by using a table.

Name of Polygon |
Polygon |
# of sides |
# of angles |
\begin{align*}n(n-3) \div 2\end{align*} |
\begin{align*}180^{\circ} (n-2)\end{align*} |

Triangle | Tri 3 | 3 | 0 | 180° | |

Square/Rectangle | 4 | 4 | 2 | 360° | |

Pentagon | Penta 5 | 5 | 5 | 540° | |

Hexagon | Hexa 6 | 6 | 9 | 720° | |

Heptagon | Hepta 7 | 7 | 14 | 900° | |

Octagon | Octa 8 | 8 | 20 | 1080° | |

Nonagon | Nona 9 | 9 | 27 | 1260° | |

Decagon | Deca 10 | 10 | 35 | 1440° |

### Guided Practice

What is the measure of each interior angle in a regular octagon?

First, write the formula to calculate the sum of the interior angles of an octagon.

\begin{align*}\text{Sum of interior angles} = 180^{\circ} (n-2)\end{align*}

Next, substitute 8 for the value of \begin{align*}n\end{align*}.

\begin{align*}\text{Sum of interior angles} = 180^{\circ} (8-2)\end{align*}

Next, perform the subtraction inside the parenthesis.

\begin{align*}\text{Sum of interior angles} = 180^{\circ}(6)\end{align*}

Then, perform the multiplication.

\begin{align*}\text{Sum of interior angles} = 1080^{\circ}\end{align*}

The answer is 1080.

The sum of the 8 interior angles is 1080°.

The interior angles of a regular octagon are congruent. The measure of each interior angle can be calculate by dividing the sum of the interior angles by the number of angles.

\begin{align*}\frac{1080^{\circ}}{8} = 135^{\circ}\end{align*}

### Examples

#### Example 1

How many diagonals can be drawn in a regular nonagon?

First, write the formula for calculating the number of diagonals for an \begin{align*}n\end{align*}-sided polygon.

\begin{align*}\# \ \text{of diagonals} = [n (n-3)] \div 2 \quad \text{Where} \ n = 9 \ \text{for a nonagon}\end{align*}

Next, substitute 9 for \begin{align*}n\end{align*} in the formula.

\begin{align*}\# \ \text{of diagonal} = [9(9-3)] \div 2\end{align*}

Next, perform the subtraction inside the parenthesis.

\begin{align*}\# \ \text{of diagonal} = [9(6)] \div 2\end{align*}

Next, perform the multiplication inside the square brackets.

\begin{align*}\# \ \text{of diagonals} = 54 \div 2\end{align*}

Then, perform the indicate division.

\begin{align*}\# \ \text{of diagonals} = 27\end{align*}

The answer is 27.

Twenty-seven diagonals can be drawn in a regular nonagon.

#### Example 2

If a dodecagon is a regular polygon with 12 sides, what is the sum of the interior angles?

First, write the formula to calculate the sum of the interior angles of a dodecagon.

\begin{align*}\text{Sum of interior angles} = 180^{\circ} (n-2)\end{align*}

Next, substitute 12 for the value of \begin{align*}n\end{align*}.

\begin{align*}\text{Sum of interior angles} = 180^{\circ} (12-2)\end{align*}

Next, perform the subtraction inside the parenthesis.

\begin{align*}\text{Sum of interior angles} = 180^{\circ}(10)\end{align*}

Then, perform the multiplication.

\begin{align*}\text{Sum of interior angles} = 1800^{\circ}\end{align*}

The answer is 1800.

The sum of the 12 interior angles is 1800°.

#### Example 3

If an icosagon is a regular polygon with 20 sides, what is the number of diagonals that can be drawn inside this polygon?

First, write the formula for calculating the number of diagonals for an \begin{align*}n\end{align*}-sided polygon.

\begin{align*}\# \ \text{of diagonal} = [n(n-3)] \div 2 \quad \text{Where} \ n = 20 \ \text{for an icosagon}\end{align*}

Next, substitute 20 for \begin{align*}n\end{align*} in the formula.

\begin{align*}\# \ \text{of diagonal} = [20 (20-3)] \div 2\end{align*}

Next, perform the subtraction inside the parenthesis.

\begin{align*}\# \ \text{of diagonal} = [20(17)] \div 2\end{align*}

Next, perform the multiplication inside the square brackets.

\begin{align*}\# \ \text{of diagonal} = 340 \div 2\end{align*}

Then, perform the indicate division.

\begin{align*}\# \ \text{of diagonal} = 170\end{align*}

The answer is 170.

One hundred and seventy diagonals can be drawn in a regular icosagon.

### Follow Up

Remember Dylan and his geodesic dome? He needs to figure out the sum of the interior angles for both a pentagon and a hexagon.

Dylan can use the formula:

\begin{align*}\begin{array}{rcl} \text{Sum of interior angles} &=& 180^{\circ} (n-2) \\ \text{Sum of interior angles} &=& 180^{\circ} (n-2) \qquad \qquad \quad \ \text{Where the value of} \ n \ \text{for a pentagon is} \ 5. \\ \text{Sum of interior angles} &=& 180^{\circ} (5-2) \qquad \qquad \quad \ \text{Perform the subtraction inside the parenthesis}. \\ \text{Sum of interior angles} &=& 180^{\circ} (3) \qquad \qquad \qquad \quad \text{Perform the multiplication} \\ \text{Sum of interior angles} &=& 540^{\circ} \end{array} \end{align*}

\begin{align*}\begin{array}{rcl} \text{Sum of interior angles} &=& 180^{\circ} (n-2) \qquad \qquad \ \ \ \ \text{Where the value of} \ n \ \text{for a hexagon is} \ 6. \\ \text{Sum of interior angles} &=& 180^{\circ} (6-2) \qquad \qquad \quad \ \text{Perform the subtraction inside the parenthesis}. \\ \text{Sum of interior angles} &=& 180^{\circ} (4) \qquad \qquad \qquad \ \ \ \text{Perform the multiplication} \\ \text{Sum of interior angles} &=& 720^{\circ} \end{array}\end{align*}

Dylan will figure out that the sum of the interior angles of a pentagon is 540° and the sum of the interior angles of a hexagon is 720°.

### Video Review

https://www.youtube.com/watch?v=qG3HnRccrQU

### Explore More

Answer true or false to each question about regular and irregular polygons.

1. The angles of a regular polygon are all the same size.

2. A regular hexagon has six sides that are different lengths.

3. An irregular pentagon has sides that are the same length.

4. An irregular polygon is one that has one side open.

5. A regular triangle could also be called an equilateral triangle.

6. The side lengths of a regular octagon are all the same length.

Identify each figure as regular or irregular. Then identify the type of polygon that it is too.

7.

8.

9. An eight sided figure with sides of equal length.

10.

11.

12.

Use the formula \begin{align*} (n-2) \times 180\end{align*} to figure out the sum of the angle measures of each polygon.

13. Regular hexagon

14. Regular octagon

15. Triangle

16. Trapezoid

17. Decagon