## Math 8 Chapter 2 Lesson 1: Polygon – Regular Polygon

## 1. Theoretical Summary

### 1.1. Polygon concept

**Define: **A convex polygon is a polygon that always lies in a half-plane whose border is a line containing any side of the polygon.

### 1.2. Polygons

**Define: **A regular polygon is a polygon with all sides equal and all angles equal.

– The sum of the measures of the angles of a polygon with n sides is \(\left( {n – 2} \right){.180^o}\)

– The measure of an angle of a regular polygon \(n\) side is \(\dfrac{{\left( {n – 2} \right){{.180}^0}}}{n}\).

– The number of diagonals of a polygon with n sides is \(\dfrac{{n\left( {n – 3} \right)}}{2}\).

## 2. Illustrated exercise

### 2.1. Exercise 1

Prove that the measure of the angle of a regular n-gon is \(\dfrac{{(n – 2){{.180}^0}}}{n}.\)

**Solution guide**

Draw an n – convex triangle, draw diagonals from a vertex of n – convex triangle, then divide that polygon into \((n – 2 )\) triangles

The sum of the angles of n – convex triangle is equal to the sum of the angles of \((n – 2)\) triangle, that is, the measure is \((n – 2 ).180^0\)

Since n – equilateral triangles have n equal angles, the measure of each angle is \(\dfrac{{(n – 2){{.180}^0}}}{n}.\)

### 2.2. Exercise 2

Calculate the measure of \(8\) regular side, \(10\) regular side, \(12\) regular side.

**Solution guide**

Applying the formula to calculate the measure of a regular n-gon is \(\dfrac{{(n – 2){{.180}^0}}}{n}\), we have:

The measure of the angle of a regular octagon is: \(\dfrac{{(n – 2){{.180}^0}}}{n}\) \(=\dfrac{{(8 – 2){{ .180}^0}}}{8}\) \(=135^0\)

The measure of the angle of a regular 10-sided figure is: \(\dfrac{{(n – 2){{.180}^0}}}{n}\) \(=\dfrac{{(10 – 2){{ .180}^0}}}{10}\) \(=144^0\)

The measure of the angle of a regular 12-sided figure is: \(\dfrac{{(n – 2){{.180}^0}}}{n}\) \(=\dfrac{{(12 – 2){{ .180}^0}}}{12}\) \(=150^0\)

### 2.3. Exercise 3

Find the number of diagonals of the shape \(8\) side, \(10\) side, \(12\) side.

**Solution guide**

Applying the formula to calculate the number of diagonals of an n-triangle is \(\dfrac{{n.(n – 3)}}{2},\) we have:

The number of diagonals of the figure \(8\) side is:

\(\dfrac{{n.(n – 3)}}{2}\) \(=\dfrac{{8.(8 – 3)}}{2}\) \(=20\) (diagonal). )

The number of diagonals of the shape \(10\) side is:

\(\dfrac{{n.(n – 3)}}{2}\) \(=\dfrac{{10.(10 – 3)}}{2}\) \(=35\) (diagonal diagonal). )

The number of diagonals of the figure \(12\) side is:

\(\dfrac{{n.(n – 3)}}{2}\) \(=\dfrac{{12.(12 – 3)}}{2}\) \(=54\) (diagonal). )

## 3. Practice

### 3.1. Essay exercises

**Question 1: **

a) Draw a figure and calculate the number of diagonals of the pentagon and hexagon

b) Prove that the n-angle has all \(\dfrac{{n.(n – 3)}}{2}\) diagonals.

**Verse 2: **Prove that the sum of the exterior angles of a (convex) polygon has a measure of \(360°.\)

**Question 3: **Which polygon has the sum of the measures of (inside) angles equal to the sum of the measures of the exterior angles?

**Question 4: **How many acute angles does a (convex) polygon have at most?

**Question 5: **A regular polygon has the sum of all the exterior angles and an interior angle equal to \(468°.\) How many sides does the regular polygon have?

### 3.2. Multiple choice exercises

**Question 1: **Each interior angle of a regular polygon with n sides is:

A. \({\left( {n – 1} \right){{.180}^0}}\)

B. \({\left( {n – 2} \right){{.180}^0}}\)

C. \(\frac{{\left( {n – 2} \right){{.180}^0}}}{2}\)

D. \(\frac{{\left( {n – 2} \right){{.180}^0}}}{n}\)

**Verse 2: **The total number of diagonals of a convex pentagon is:

A. 7

B. 8

C. 5

D. 10

**Question 3: **Choose the correct sentence: Give the shapes: Rectangle, rhombus, square, isosceles triangle, equilateral triangle.

How many regular polygons are there in the figures above?

A. 1

B. 2

C. 3

D. 4

**Question 4: **If a polygon has 54 diagonals, the number of sides is .

A. 9

B. 10

C. 5

D. 12

**Question 5: **Given an 8-sided polygon, the number of diagonals of that polygon is

A. 40

B. 28

C. 20

D. 16

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Understand the concept of polygons, convex polygons, regular polygons. Know the sum of the measures of the angles of a polygon.
- Train students in drawing, measuring and calculating skills. Especially drawing regular polygons with its symmetry axes.

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