## Math 8 Chapter 4 Lesson 7: Regular pyramids and regular truncated pyramids

## 1. Summary of theory

### 1.1. 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.

### 1.2. Regular pyramid

A regular pyramid is a pyramid whose base is a regular polygon, and its side faces are equilateral 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.

### 1.3. Evenly 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.

The image above has a truncated image all A’B’C’D’.ABCD

Each side of the truncated pyramid is an isosceles trapezoid.

## 2. Illustrated exercise

### 2.1. Exercise 1

Determine whether the following statements are true or false?

a) A regular pyramid has a rhombus base and the base of the altitude coincides with the intersection of the two diagonals of the base.

b) A regular pyramid has a rectangular base and the base of the altitude coincides with the intersection of the two diagonals of the base.

**Solution guide**

a) Wrong, because a rhombus is not an equilateral quadrilateral (the angles are not equal).

b) Wrong, because the rectangle is not an equilateral quadrilateral (the sides are not equal).

### 2.2. Exercise 2

Look at the pictures below and fill in the appropriate phrases and numbers and in the blanks, knowing the shapes below are regular pyramids.

__Solution guide__

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given an equilateral quadrilateral pyramidal S.ABCD whose side faces are equilateral triangles AB = 8cm, O is the midpoint of AC. What is the length of segment SO?

**Verse 2:** Regular hexagonal pyramid S.ABCDEH has AB = 6cm, side SA = 10cm. What is the height of the pyramid?

**Question 3:** Calculate the total area of the equilateral triangular pyramid according to the dimensions in the figure.

**Question 4:** If a regular quadrilateral pyramid has base length = 6cm and height 4cm, the surrounding area is ?

### 3.2. Multiple choice exercises

**Question 1:** A regular pyramid S.ABCD has a base of square ABCD. Given AB=30cm, height of side triangle SH=25m. The height SO of the pyramid is: (m)

A. \(\sqrt{850}\)

B. \(15\)

C. \(20\)

D. \(\sqrt{675}\)

**Verse 2: **Let the pyramid of equilateral quadrilateral S.ABCD. Which answer is correct?

A. \(SO^{2}=SA^{2}+\frac{AB^{2}}{2}\)

B. \(SO^{2}=SA^{2}+2AB^{2}\)

C. \(SA^{2}=SO^{2}+\frac{AB^{2}}{2}\)

D. \(SA^{2}=SO^{2}+2AB^{2}\)

**Question 3:** Regular hexagonal pyramid S.ABCDEH has AB=3m, side SA=5m. The length and height of the pyramid are:

A. 4m

B. \(\sqrt{8}\)m

C. 2m

D. 8m

**Question 4: **Choose the correct statement from the following statements:

A. If a pyramid has a base of a rhombus and the base of the altitude coincides with the center of the rhombus, then it is a regular pyramid

B. If the base is a rectangle, the base of the altitude coincides with the intersection of the two diagonals of the base, then it is a regular pyramid.

C. If a pyramid has a square base, it is a regular pyramid

D. If the base of the pyramid is an equilateral triangle and the base of the altitude coincides with the center of the triangle, then it is an equilateral pyramid.

**Question 5:** Choose the correct statement from the following statements:

A. If a pyramid has a base of an equilateral quadrilateral and two sides are isosceles triangles, then it is an equilateral quadrilateral pyramid.

B. If a pyramid has a rectangular base and the base of the altitude coincides with the intersection of the two diagonals of the base, then it is a regular pyramid.

C. If a pyramid has a square base, it is a regular pyramid.

D. If the base of the pyramid is an equilateral triangle and the base of the altitude coincides with the center of the triangle, then it is an equilateral pyramid.

## 4. Conclusion

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

- Understand the concept of regular pyramids and regular truncated pyramids (top, sides, sides, bottom, height, midline)
- Know how to name a pyramid according to its base polygon and draw an equilateral triangular pyramid in four steps

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