## Math 8 Chapter 2 Lesson 6: Area of Polygon

## 1. Summary of theory

**Method for calculating the area of a polygon**

Calculating the polygonal area of any polygon is often referred to as calculating the area of triangles. We can divide polygons into triangles or create a triangle containing polygons

In some cases, to facilitate the calculation, we can divide the polygon into many right triangles and square trapezoids.

## 2. Illustrated exercise

### 2.1. Exercise 1

Make the necessary measurements (to the nearest mm) to calculate the area of \(ABCDE\) (h.\(152\)).

**Solution guide**

The polygon \(ABCDE\) is divided into triangle \(ABC\), two right triangles \(AHE, DKC\) and right triangle \(HKDE.\)

Make the measurement to the nearest mm, we get:

\(BG= 19mm, AC = 48mm, \)\(AH = 8mm, HK = 18mm\)

\(KC = 22mm, EH = 16mm, \)\(KD = 23mm\)

\({S_{ABC}}= \dfrac{1}{2}.BG.AC = \dfrac{1}{2}. 19.48 = 456\) \((m{m^2})\)

\({S_{AHE}}=\dfrac{1}{2} AH. HE = \dfrac{1}{2} 8.16 = 64\) \((m{m^2})\)

\({S_{DKC}}=\dfrac{1}{2}.KC.KD = \dfrac{1}{2}. 22.23 \)\(\,= 253\) \((m{m^2) })\)

\({S_{HKDE}}=\dfrac{\left ( HE+KD \right ).HK}{2} \)\(\,= \dfrac{\left (16+23 \right ).18}{ 2}= 351\) \((m{m^2})\)

Therefore

\({S_{ABCDE}} = {S_{ABC}} + {S_{AHE}} + {S_{DKC}} \)\(\,+ {S_{HKDE}} = 456 + 64 + 253 + 351 \)\(\,= 1124\;(m{m^2})\)

So \({S_{ABCDE}} = 1124\;(m{m^2})\)

### 2.2. Exercise 2

Make the necessary drawings and measurements to calculate the area of a land mass of the form \(154\), where \(AB // CE\) and scaled \(\dfrac{1}{5000 }\)

**Solution guide**

Divide the land mass \(ABCDE\) into a trapezoid \(ABCE\) and a triangle \(ECD.\) It is necessary to draw the altitude \(CH\) of the trapezoid and the altitude \(DK\) of the triangle. Make measurements to \(mm\) we get \(AB = 30\,mm, CE = 26\,mm,\) \(CH = 13\,mm, DK = 7\,mm.\ )

\({S_{ABCE}}=\dfrac{\left ( AB+EC \right ).CH}{2} \)\(\,= \dfrac{\left ( 30 + 26 \right ).13}{ 2} =364\) \((m{m^2})\)

\({S_{ECD}}=\dfrac{1}{2} EC. DK = \dfrac{1}{2} .26.7= 91\) \((m{m^2})\)

Thus \({S_{ABCDE}} = {S_{ABCE}} + {S_{ECD}} = 364 + 91 \)\(\,= 455\) \((m{m^2})\)

Since the map is plotted at scale \(\dfrac{1}{5000}\) the area of the land mass is:

\(S = 455. 5000 = 2275000 \;(m{m^2}) \)\(\,= 2,275 \;({m^2})\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Make the necessary drawings and measurements to calculate the area of the polygon \(ABCDE\) \((BE // CD)\) (h.189)

**Verse 2:** According to the dimensions given in figure 191, calculate the area of the striped shape (unit \(m^2\) ).

**Question 3: **Find the area of the land according to the dimensions shown in Figure 192 (unit \(m^2\))

### 3.2. Multiple choice exercises

**Question 1: **The measure of each interior and exterior angle of a regular pentagon is:

A. \(75^{\circ}; 150^{\circ}\)

B. \(108^{\circ}; 72^{\circ}\)

C. \(100^{\circ}; 80^{\circ}\)

D. \(110^{\circ}; 70^{\circ}\)

**Verse 2:** Given parallelogram ABCD with CD = 4cm, the height drawn from A to side CD is 3cm. Let M be the mid point of AB. DM intersects AC at N. Calculate area of triangle AMN

A. 4 \(cm^{2}\)

B. 10 \(cm^{2}\)

C. 2 \(cm^{2}\)

D. 1 $cm^{2}$

**Question 3: **Let ABC be a triangle with area 12 \(cm^{2}\). Let N be the midpoint of BC, on AC such that AM = \frac{1}{3}AC, AN intersects BM at O. Calculate the area of triangle AOM

A. 2 \(cm^{2}\)

B. 1 \(cm^{2}\)

C. 3 \(cm^{2}\)

D. 6 \(cm^{2}\)

**Question 4: **Given rhombus MNPQ. Knowing A, B, C, D are the midpoints of the sides NM, NP, PQ, QM, respectively. Find the ratio of the areas of quadrilateral ABCD and rhombus MNPQ.

A. \(\frac{1}{2}\)

B. \(\frac{2}{3}\)

C. 2

D. \(\frac{1}{3}\)

**Question 5:** The measure of each corner of a regular 9-sided figure is:

A. \(120^{\circ}\)

B. \(60^{\circ}\)

C. \(140^{\circ}\)

D. \(135^{\circ}\)

## 4. Conclusion

Through this lesson, you will learn some key topics as follows:

- Master simple polygonal area formulas, especially how to calculate the area of triangles and trapezoids.
- Know how to properly divide the polygon to find the area into many simple polygons.
- Know how to make the necessary drawings and measurements.

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