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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