## Math 9 Chapter 4 Lesson 3: Sphere Sphere Area and Sphere Volume

## 1. Summary of theory

### 1.1. Globular

When rotating a semicircle with center \(O\), radius \(R\) one circle around a fixed diameter \(AB\), a sphere is obtained.

– The point \(O\) is called the center, the length \(R\) is the radius of the sphere.

– The semicircle in the above rotation makes up the sphere

### 1.2. Area of the bridge surface

Recalling the knowledge learned in the lower class, we have the following formula:

\(S=4\pi R^2=\pi d^2\) (where R is the radius, d is the diameter of the sphere)

### 1.3. Volume of sphere

The formula for calculating the volume of a sphere:

\(V=\frac{4}{3}\pi R^3\)

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Find the area of a sphere with radius \(4cm\)

**Solution guide**

According to the formula, the area of the sphere is: \(S=4R^2 \pi=4.4^2\pi=64\pi(cm^2)\)

**Verse 2: **When the radius of a sphere is increased by \(\frac{3}{2}\) times, how does its area and volume change?

**Solution guide**

According to the formula for calculating the area, we have \(S=4R^2 \pi\)

Based on the above formula, as R increases \(\frac{3}{2}\) the area increases \(R^2\) times i.e. \(\frac{9}{4}\) times .

Same for volume: \(V= \frac{4}{3}\pi R^3\)

when R increases \(\frac{3}{2}\) then the volume increases \(R^3\) times i.e. \(\frac{27}{8}\) times.

**Question 3: **Assuming an orange has a similar shape to a sphere, Lan cuts the orange in half and measures the diameter of the half of the orange that has just been cut, you can measure the radius including the orange peel as \(2.5cm\), knowing the peel thick orange \(3mm\). Calculate the net volume of oranges that Lan ate.

**Solution guide**

Consider the orange part that Lan ate is also a type of sphere, so the radius of that sphere is the radius Lan measured minus the shell.

That is: \(R=25-3=22mm\)

So, the volume of oranges that Lan ate is: \(V= \frac{4}{3}\pi R^3=\frac{4}{3}\pi. 22^3=\frac{42592\ pi}{3}(mm^3)\)

### 2.2. Advanced exercises

**Question 1: **Calculate the radius of a sphere, knowing that the sphere has an algebraic measure of area equal to an algebraic measure of volume.

**Solution guide**

According to the title, we have:

\(\frac{4}{3}\pi R^3=4R^2.\pi\Leftrightarrow \frac{R}{3}=1\Leftrightarrow R=3(dvdd)\)

**Verse 2: **Calculate the volume of a sphere with radius (cm) satisfying the equation: \(x^2-3x-4=0\)

**Solution guide**

Solving the above equation, we get two solutions with opposite signs, choose only the solution \(x=4(cm)\)

So, the volume of that sphere is \(V=\frac{4}{3}\pi R^3=\frac{4}{3}.4^3.\pi=\frac{256\pi} {3}(cm^3)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **An equilateral triangle ABC has side length a, circumscribed a circle. Given the figure rotates around the altitude AH of that triangle (see figure), we get a cone circumscribing a sphere.

Calculate the volume of the cone outside the sphere

**Verse 2: **A spherical ball inside a cube as shown below:

a. Calculate the ratio of the total area of the cube to the area of the sphere

b. If the area of the sphere is 7π (cm2), what is the total area of the cube?

c. If the radius of the sphere is 4cm, what is the volume of the empty part (in the box outside the sphere)?

**Question 3: **Pick a watermelon: with two watermelons (as two spheres) one large and one small, the ratio of their diameters is 5:4 but the price of the large one is one-and-a-half times the price of the small one. Which fruit do you choose to buy more profitable (see their quality is the same)?

**Question 4: **With a tape measure, is it possible to determine the volume of a spherical object?

### 3.2. Multiple choice exercises

**Question 1: **Which of the following figures has the largest area?

(A) The circle has radius \(2cm\).

(B) A square has side length \(3.5cm\).

(C) Triangle with side lengths \(3cm, 4cm, 5cm\).

(D) Hemisphere radius \(4cm\).

**Verse 2: **The two spheres \(A\) and \(B\) have radii \(x\) and \(2x\; (cm)\).

The ratio of the volumes of this sphere \(2\) is:

(A) \(1: 2\) (B) \(1: 4\) (C) \(1: 8\) (D) Another result.

**Question 3:** Fill a conical measuring device with water and then pour all the water into a cylinder with a base radius equal to the base radius of the cone and a height equal to the height of the cone. Repeat this process. until the cylinder is filled with water, the number of times to fill the cone is:

A.1 B.2 C.3 D.4

**Question 4: **Choose the wrong sentence

A. The plane that cuts through the center of the sphere is the circle with the largest area

B. The plane that cuts through the center of the sphere is the circle with the greatest circumference

C. At 2 points outside the sphere, we can always cut the circle with the largest area

D. At 3 points outside the sphere, we can always cut the circle with the largest area

**Question 5: **The volume of a sphere is \(\frac{500\pi}{3} (cm^3)\), the radius of that sphere is:

A. \(R=5cm\)

B. \(R=6cm\)

C. \(R=7cm\)

D. \(R=8cm\)

## 4. Conclusion

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

- Consolidate the concepts of the sphere, the formula for calculating the area of a sphere.
- Understand how to form the formula for the volume of a sphere and know how to apply the formulas to exercises.

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