## Math 8 Chapter 4 Lesson 5: Surrounding area of a vertical prism

## 1. Theoretical Summary

### 1.1. Surrounding area

The surrounding area of a vertical prism is equal to the sum of the areas of the sides or the perimeter of the base times the height.

\({S_{xq}} = 2p.h\)

\(p\) is the half circumference of the base, \(h\) is the height

### 1.2. Total area

The total area of the prism is equal to the sum of the surrounding areas and the areas of the two bases.

## 2. Illustrated exercise

### 2.1. Exercise 1

Calculate the surrounding area, the total area of the following vertical prisms (h.102):

**Solution guide**

a) With the figure on the left:

The surrounding area of the vertical prism is: \(2.(3+ 4) . 5 = 70 (cm^2) \)

The total area of the vertical prism is: \(70 + 2.3.4. = 94(cm^2) \)

b) With the figure on the right:

\( \triangle ABC \) square at \(A \Rightarrow BC^2 = AB^2 + AC^2 = 9 + 4 = 13\)

\( \Rightarrow BC = \sqrt{13} (cm) \)

The bottom half circumference is: \(\dfrac{2+3+\sqrt {13}}{2}=\dfrac{5+\sqrt {13}}{2}\)

The surrounding area of the vertical prism is: \( 2.\dfrac{5+\sqrt {13}}{2}.5 = 25 + 5\sqrt{13} (cm^2 )\)

The total area of the vertical prism is: \( 25 + 5\sqrt{13} + 2(\dfrac{1}{2}. 2.3) \) \(= 25 + 5\sqrt{13} + 6= 31 + 5\sqrt{13}(cm^2 ) \)

### 2.2. Exercise 2

a) From the expansion figure (h.105) is it possible to fold along the edges to get a vertical prismatic plate? (The quadrilaterals in the figure are all rectangles).

b) In the folded figures, consider the following statements, which statement is correct?

– Side \(AD\) is perpendicular to side \(AB\).

– \(EF\) and \(CF\) are two sides that are perpendicular to each other.

– Side \(DE\) and side \(BC\) are perpendicular to each other.

– The two bases \((ABC)\) and \((DEF)\) lie on two parallel planes.

– The plane \((ABC)\) is parallel to the plane \((ACFD)\).

**Solution guide**

a) From the lateral expansion, we can fold along the edges to get a vertical prism.

b) The statements are true:

– Side \(AD\) is perpendicular to side \(AB\).

– \(EF\) and \(CF\) are two sides that are perpendicular to each other.

– The two bases \((ABC)\) and \((DEF)\) lie on two parallel planes.

## 3. Practice

**Question 1: **What is the total area of the vertical prism wall cabinet shown in the figure?

**Verse 2: **People cut a block of wood in the shape of a cube as shown in figure 124 (cut in the face \((AC{C_1}{A_1})\) and get two vertical prisms).

a) Is the base of the obtained vertical prism a right triangle, isosceles triangle, or an equilateral triangle?

b) Are the sides of each obtained vertical prism all squares?

**Question 3: **A vertical prismatic glass paperweight has the dimensions shown in figure 126. Its total area is:

**Question 4: **

The base of the correct prism is an isosceles trapezoid with sides b = 11 mm, a = 15 mm and height hr = 7 mm. The height of the vertical prism is h = 14 mm. Calculate the area around the prism.

### 3.2. Multiple choice exercises

**Question 1: **The formula \(S_{xq}=2p.h\), where p is the half circumference of the base, h is the height, which is the formula for calculating its surrounding area:

A. Vertical prism

B. Rectangular shape

C. Cube

D. All 3 sentences are correct

**Verse 2: **A vertical prism-shaped wooden block has a height of 4m, the base is a convex quadrilateral ABCD. Know AC=90cm. The perpendicular segment from B to AC is 30cm, and the perpendicular segment from D to AC is 20cm.Know that each square meter of this wood is 4 million VND. The price of this block is:

A. 2 million dong

B. 3 million dong

C. 3 million 6 hundred thousand dong

D. 2 million 8 hundred thousand dong

**Question 3:** Given a triangular prism ABC.A’B’C’ with base perimeter is 4.5cm, surrounding area is \(18cm^{2}\). Choose the correct statement from the following statements:

A. AA’=CC’>BB’

B. AA’=4cm

C. CC’=9cm

D. BB’>4cm

**Question 4: **Let ABC be a triangular vertical prism. A’B’C’ whose side faces are rectangles of equal area. The height of the prism is 6 m, and the base side of the prism is 4 m. Area The surroundings of the prism are:

A. \(96m^{2}\)

B. \(36m^{2}\)

C. \(72m^{2}\)

D. \(144m^{2}\)

**Question 5:** Calculate the total area of a vertical prism ABC.A’B’C’, base is a right triangle at A, according to the following dimensions: AB=6cm, BC=10cm, CC’=15cm.

A. \(408cm^{2}\)

B. \(360cm^{2}\)

C. \(900cm^{2}\)

D. \(204cm^{2}\)

## 4. Conclusion

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

- Understand how to calculate the perimeter of a vertical prism.
- Know how to apply formulas to calculations with specific shapes.

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