## Math 8 Review chapter 4: Vertical prism. Regular pyramid

## 1. Theoretical Summary

### 1.1. Rectangular

**Define: **A rectangular box is a space shape with 6 faces that are all rectangles.

A rectangular box has 6 faces, 8 vertices, and 12 edges.

The two faces facing each other are considered to be the base of the rectangular box, the other faces are called the side

A cube is a rectangular box with 6 faces that are all squares.

**a) The volume of the rectangular box**

We have V = abh

**b) Can like cubes**

We have: V = a^{3}.

### 1.2. Plane and line

- Through three noncollinear points define one and only one plane.
- Through two intersecting lines define one and only one plane.
- If a line passes through two distinct points of a plane, then every point of that line is in the plane.

### 1.3. Two parallel lines in space

– Two lines a and b are said to be parallel if they lie in the same plane and have no points in common. Symbol a // b.

– Two distinct lines parallel to a third line are parallel to each other.

**Attention: **Two distinct lines in space can:

– Intersect – Parallel – Diagonal (not in the same plane)

### 1.4. The line is parallel to the plane. Two parallel planes

**a) A line parallel to the plane**

– A line a is said to be parallel to a plane ( P ) if the line is not in the plane ( P ) and is parallel to a line d in the plane.

Sign a // ( P ).

– If a line is parallel to a plane, they have no point in common.

**b) Two parallel planes**

– If the plane ( Q ) contains two intersecting lines parallel to the plane ( P ), then the plane ( Q ) is parallel to the plane ( P ). Symbol ( Q )//( P ).

Two planes that are parallel to each other have no point in common.

– Two distinct planes have a common point, then they share a line passing through that common point (that common line is called the intersection of the two planes).

### 1.5. The line is perpendicular to the line. Two perpendicular planes

**a) The line is perpendicular to the plane**

– The line d is said to be perpendicular to the plane ( P ) if the line d is perpendicular to two intersecting lines in the plane ( P ). The symbol d ⊥ ( P ).

– If a line a is perpendicular to the plane ( P ) at point A then it is perpendicular to every line inside ( P ) and passing through point A.

**b) Two perpendicular planes**

– The plane ( P ) is said to be perpendicular to the plane ( Q ) if the plane ( P ) contains a line perpendicular to the plane ( Q ). The symbol ( Q ) ⊥ ( P ).

### 1.6. Vertical prism

– Two bases are two congruent polygons and lie on two parallel planes.

– The sides are parallel, equal and perpendicular to the two base planes. The side length is called the height of the vertical prism.

The side faces are rectangles and are perpendicular to the two base planes.

– Rectangular boxes, cubes are vertical prisms.

A vertical prism whose base is a parallelogram is called a vertical box.

### 1.7. Area – Volume of a vertical prism

**a) Formula for surrounding area**

The surrounding area of a vertical prism is equal to the perimeter of the base times the height:

S_{xq }= 2p.h (p: half circumference of base, h: height)

**b) Total area**

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

S_{city} = WILL_{xq} + 2S (S: bottom charge)

**c) Volume**

The volume of a vertical prism is equal to the area of the base times the height:

V = Sh (S: base area, h: height)

### 1.8. Pyramid

– The base is a polygon, the sides are triangles that share a vertex.

The line that passes through the top and is perpendicular to the plane of the base is called the altitude.

### 1.9. Regular pyramid

A regular pyramid is a pyramid whose base is a regular polygon, and its side faces are isosceles triangles with a common vertex.

+ The base of the height of the pyramid coincides with the center of the circle passing through the vertices of the base face.

+ The altitude drawn from the top of each side face of the pyramid is called the midpoint of that pyramid.

### 1.10. Regular truncated pyramid

A regular truncated cone is the part of a regular pyramid that lies between the plane of the base of the pyramid and the plane parallel to the base and intersects the pyramid.

Each side of the truncated pyramid is an isosceles trapezoid.

### 1.11. Area – Volume of a regular pyramid

**a) Surrounding area of a regular chop**

The perimeter of a regular pyramid is equal to the product of the semicircle of the base and the midpoint:

Sxq = pd (p: half circumference of base, d: midpoint)

**b) Total area of the pyramid**

The total area of the pyramid is equal to the sum of the surrounding area and the area of the base:

Stp = Sxq + S (S: area of bottom)

**c) Formula for the volume of a regular pyramid**

The volume of the pyramid is one third of the area of the base times the height:

V = \(\frac{1}{3}\).Sh (S: area of base, h: height)

## 2. Illustrated exercise

### 2.1. Exercise 1

Calculate the perimeter, total area and volume of a vertical prism with height \(h\) and base with a square of side \(a\);

**Solution guide**

p is the half circumference of the base and h is the height of the prism.

The surrounding area is:

\({S_{xq}} = 2p.h = 4.a. {\text{ }}h\)

The area of a base is:

\({S_đ} = {a^2}\)

The total area of the vertical prism is:

\({S_{tp}} = {S_{xq}} + 2{S_đ} = 4ah + 2{a^2}\)

Prism volume:

\(V = {S_đ}h = {a^2}.h\)

### 2.2. Exercise 2

Calculate the total area of the bar as shown in Figure 142 (the front and back sides of the bar are isosceles trapezoids, the other four faces are rectangles, given (\(\sqrt {10} \approx). 3.16\)).

**Solution guide**

The wooden bar is in the shape of a vertical prism, the bottom is an isosceles trapezoid. We find the height of an isosceles trapezoid. We have:

\(\eqalign{

& DH = {1 \over 2}\left( {DC – AB} \right) \cr

& \,\,\,\,\,\,\,\,\,\,\,\, = {1 \over 2}\left( {6 – 3} \right) = 1.5\left( {cm} \right) \cr} \)

Height:

\(AH =\sqrt {AD^2 – DH^2}= \sqrt {3,{5^2} – 1,{5^2}} \)\(\,= \sqrt {12,25 – 2 ,25} \) \(= \sqrt {10} \approx 3.16\left( {cm} \right)\)

The area around the prism is:

\({S_{xq}}= 2ph = (3 + 6 + 3.5 + 3.5).11.5\)\(\,=16.11.5 = 184 \,(cm^2)\)

The area of one bottom surface is:

\(S_đ= \dfrac{{\left( {3 + 6} \right).3,16}}{2} = 14,22c{m^2}\)

Total area:

\({S_{tp}} = {S_{xq}} + 2{S_đ}\)\(\,= 184 + 2.14.22 = 212.44 \,(cm^2)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **The water tank is in the shape of a vertical prism with the dimensions shown in the figure.

a.Calculate the surface area of the tub (excluding the lid).

b. Calculate the volume of the tub.

c. When the tank is full of water, how much can it hold?

d. How much paint is needed to paint both the inside and outside of the tub (one liter of paint covers 16 meters).

f. How long does it take for a pump with a capacity of 125 liters/min to pump a quantity of water into the tank to a height of 1.05m from the bottom of the tank?

**Verse 2: **Calculate the total area of the vertical prism according to the dimensions given in the figure.

**Question 3: **The trunk of a truck has the shape of a vertical prism, the dimensions are given above

a) Calculate the volume of the container.

b) If \(1m^3\) sand weighs \(1,6\) tons and the truck is loaded with \(\displaystyle {3 \over 4}\) its payload, what is the weight of the sand?

c) When the sand is leveled and filled, what is its area inside the truck?

### 3.2. Multiple choice exercises

**Question 1:** A regular quadrilateral pyramid has a height of 35 cm and a base 24 cm. Calculate the median length.

A. 37 cm

B. 73 cm

C. 27 cm

D. 57 cm

**Verse 2:** Let ABCD.A′B′C′D′ cube with \(A′C = \sqrt{3}\). Calculate the volume of a cube.

A. \(3a^{3}\sqrt{3}\)

B. \(a^{3}\)

C. \(27a^{3}\)

D. \(9a^{3}\)

**Question 3: **ABCD rectangular box. A′B′C′D′ whose base ABCD is a square with side a and the area of rectangle ADC′B′ equals \(2a^{2}\), what is the surrounding area of the rectangular box?

A. WILL_{xq} = \(4a^{2}\sqrt{3}\)

B. WILL_{xq} = \(2a^{2}\sqrt{3}\)

C. WILL_{xq} = \(4a^{2}\)

D. WILL_{xq} = \(4a^{2}\sqrt{2}\)

**Question 4:** Let ABCD.A’B’C’D’ rectangular box have AB = 2cm, AD = 3cm, AA’ = 4cm. Volume of rectangular box ABCD.A’B’C’D’ ?

A. 12 \(cm^{3}\)

B. 24 \(cm^{3}\)

C. 18 \(cm^{3}\)

D. 15 \(cm^{3}\)

**Question 5:** Let ABCD.A′B′C′D′ rectangular box. Find the area of rectangle ADC′B′ knowing AB = 28cm, \(B′D^{2}\) = 3709, DD′ = 45cm.

A. 1950 \(cm^{2}\)

B. 206 \(cm^{2}\)

C. 1509 \(cm^{2}\)

D. 1590 \(cm^{2}\)

## 4. Conclusion

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

- Grasp the concept of a rectangular box and a line, two parallel lines in space.
- Understand the elements of a rectangular box, know how to determine the number of faces, vertices, and sides of a rectangular box.

.

=============

## Leave a Reply