## Math 8 Chapter 4 Lesson 9: Volume of a regular pyramid

## 1. Theoretical Summary

Formula for volume:

The volume of a regular pyramid is one-third the area of the base times the height

\(V = \dfrac{1}{3} .Sh\)

\(S\): bottom area

\(h\): height

## 2. Illustrated exercise

### 2.1. Exercise 1

Figure 129 is a student summer camp tent with dimensions.

a) What is the volume of air inside the tent?

b) Determine the amount of canvas needed to pitch the tent (doesn’t take into account contours, folds, … know \(\sqrt{5} 2.24\)).

**Solution guide**

a) The required volume is equal to the volume of a pyramid with height \(2cm\), the base is a square with side length \(2m\).

The area of the base is: \( S_{đ} = 2.2=4(m^2)\)

The volume of the pyramid is: \(V = \dfrac{1}{3}.Sh = \dfrac{1}{3}.4.2 = \dfrac{8}{3}\approx 2.67\)\(\ ,(m^3) \)

b) The number of canvas to be calculated is the area of four faces (or the surrounding area) each side is an isosceles triangle.

Let \(H\) be the midpoint of \(CD\) and \(O\) the center of the square \(ABCD\).

To calculate the surrounding area, we need to calculate the midpoint ie the altitude \(SH\) of each face.

According to the Pythagorean theorem in right triangle SHA, we have:

\(SH^2 =SO^2+OH^2 \)\(\,= SO^2+{\left( {\dfrac{{BC}}{2}} \right)^2} \) \( = 2^2+1^2=5\)

\( \Rightarrow SH =\sqrt{5}\approx 2.24(m) \)

The perimeter of the pyramid is:

\( S_{xq} = pd = \dfrac{1}{2}. 2.4.2.24 = 8.96 (m^2) \)

### 2.2. Exercise 2

Calculate the total area of a regular quadrilateral pyramid, knowing the base side \(a = 5cm\), the side \(b = 5cm,\;\sqrt{18.75}\approx 4,33 \)

**Solution guide**

From the problem we have that the sides of the pyramid are equilateral triangles of side \(5cm\).

The altitude of each side is:

\(d=SH = \sqrt{SC^{2} -HC^{2}}\)

\(= \sqrt{5^{2} -2.5^{2}}= \sqrt{18.75}\approx 4.33 (cm) \)

The area around the pyramid is:

\(S_{xq} = pd = \dfrac{1}{2}. 5.4.4,33 = 43.3 (cm^2) \)

Area of the base of the pyramid: \(S_{đ} = a^2 = 5^2 =25(cm^2) \)

Total area of the pyramid:

\( S_{tp} = S_{xq}+ S_{đ} = 43.3 + 25 = 68.3 \) \((cm^2)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **A laboratory planter has the shape of a vertical prism with dimensions as shown in the figure where EDC is an isosceles triangle. Calculate :

a) Area of figure ABCDE

b) Calculate the volume of the greenhouse

c) Area of glass required to “roof” two roofs and four walls of the house

**Verse 2: **The pyramid of Cheop (25th century BC) is a regular quadrilateral pyramid, the base is equal to \(233m\), the height of the pyramid \(146.5m.\)

a) What is the side length?

b) Calculate the perimeter of the pyramid.

c) Calculate the volume of the pyramid.

**Question 3: **Pyramid of Luvre (Louvre) (Built in 1988).

People make a model of a pyramid at the entrance of the Louvre museum (France). The model has the shape of a regular pyramid with a height of \(21m,\) and the length of the bottom edge is \(34m.\)

a) What is the side edge of the pyramid?

b) Calculate the volume of the pyramid.

c) Calculate the total area of the glass panels to cover this pyramid \(({S_{xq}})\).

**Question 4:** What is the volume of a regular pyramid given the dimensions in the figure?

### 3.2. Multiple choice exercises

**Question 1:** The volume of a pyramid whose base is an equilateral triangle with sides of 6 and sides equal to \(\sqrt{15}\) is:

A. 6

B. 9

C. \(\frac{27}{2}\)

D. \(\frac{9\sqrt{3}}{2}\)

**Verse 2: **A regular quadrilateral pyramid has side length 5cm and pyramid height 4cm. The volume of the pyramid is:

A. 30

B. 24

C. 22

D. 18

**Question 3:** A regular quadrilateral pyramid and a regular quadrilateral vertical prism are shown in the figure below (base and height are equal). If the volume of the pyramid is V, then the volume of the prism is:

A. VU

B. 2V

C. 3V

D. 4V

**Question 4:** For each vertex of the cube, we consider the pyramid defined by that vertex and the midpoints of the three sides that come from that vertex. When we cut out these pyramids, the ratio of the remaining volume to the volume of the cube. Which of the following is closest to the original cube?

A. 75%

B. 68%

C. 81%

D. 84%

**Question 5: **A figure consisting of two conjoined pyramids is formed by joining the centers of the consecutive faces of the cube. so the ratio of that generated volume to the cube:

A. \(\frac{\sqrt{3}}{12}\)

B. \(\frac{\sqrt{6}}{16}\)

C. \(\frac{1}{6}\)

D. \(\frac{\sqrt{2}}{8}\)

## 4. Conclusion

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

- Know how to calculate the volume of a regular pyramid.
- Know how to apply calculation formulas to specific shapes.

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