## Math 9 Chapter 3 Lesson 10: Area of a circle, a circular fan

## 1. Summary of theory

### 1.1. Formula for calculating the area of a circle

The area of a circle with radius R is calculated by the formula: \(S=\pi R^2\)

### 1.2. Formula to calculate the area of a circular fan

The area of a circular fan of radius R, arc n0 is calculated by the formula

\(S=\frac{\pi R^2n}{180}\) or \(S=\frac{l R}{2}\) (where \(l\) is the arc length n0 of the circular fan )

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1:** Find the area of the circumcircle of a square with side 10cm

**Solution guide**

We already know that the radius of the circumcircle of square ABCD is OA.

Side of square is 10cm so AB=10cm

Applying Pythagorean theorem to right triangle OAB we have \(2OA^2=AB^2\Rightarrow2OA^2=10^2\Rightarrow OA=5\sqrt{2} (cm)\)

The area of the circle circumscribed about the square is \(S=\pi R^2=\pi (5\sqrt{2})^2=50\pi (cm^2)\)

**Verse 2:** Find the area of a circular fan whose radius is 20cm and whose arc measure is 300

**Solution guide**

Applying the formula for calculating the area of a circular fan, we have

\(S=\frac{\pi R^2n}{360}=\frac{\pi.20^2.30}{360}=\frac{100\pi}{3} (cm^2)\)

Question 3: How will the area of a circle change if the radius is tripled?

**Solution guide**

We have \(S_1=\pi R^2\) where R is the circumferential radius

After increasing the radius 3 times, the new circle radius is 3R

Then the area of the circle is \(S_2=\pi (3R)^2=9\pi R^2\)

Since \(\frac{S_2}{S_1}=\frac{9\pi R^2}{\pi R^2}=9\) the area of the circle increases 9 times after changing the radius

### 2.2. Advanced exercises

The circumference of a circle is \(16\pi (cm)\). Find the area of a circular fan whose arc measure is 50^{0}

**Solution guide**

The circumference of the circle is \(16\pi (cm)\) so \(C=2\pi R\Rightarrow R=\frac{16\pi}{2\pi}=8(cm)\)

The area of a circular fan with the measure of arc 500 is \(S=\frac{\pi R^2n}{360}=\frac{\pi 8^2.50}{360}=\frac{80}{9}\pi (cm^2)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **\(a)\) Fill in the blanks in the following table (\(S\) is the area of a circle with radius \(R\)).

\(R\) |
\(0\) |
\(first\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
\(ten\) |
\(20\) |

\(S\) |

\(b)\) Draw a graph representing the area of a circle against its radius.

\(c)\) Is the area of a circle proportional to the radius?

**Verse 2: **\(a)\) Fill in the blanks in the following table (\(S\) is the area of \(n^\circ\)).

The arc \(n^\circ\) |
\(0\) |
\(45\) |
\(90\) |
\(180\) |
\(360\) |

\(S\) |

\(b)\) Draw a graph representing the area of the fan in terms of \(n^\circ\).

\(c)\) Is the area of a fan proportional to the measure of degrees of arc\(?\)

**Question 3:** Given the interior line \((O; R).\) Divide this circle into three arcs whose measure is proportional to \(3, 4\) and \(5\) and then calculate the area of the cones formed .

**Question 4: **Let \(ABC\) triangle inscribed in circle \((O; R)\) with \(\widehat C = {45^\circ}\).

\(a)\) Calculate the area of a circular fan \(AOB\) (corresponding to minor arc \(AB\))

\(b)\) Calculate the area of \(AmB\) (corresponding to minor arc \(AB\))

### 3.2. Multiple choice exercises

**Question 1: **Find the area of a circular fan with radius 10cm and arc length \(\frac{5}{2}\pi (cm)\)

A. \(50\pi (cm^2)\)

B. \(\frac{25}{3}\pi (cm^2)\)

C. \(\frac{25}{2}\pi (cm^2)\)

D. \(25\pi (cm^2)\)

**Verse 2:** The area of a semicircle of diameter 4R is

A. \(R^2\pi\)

B. \(4R^2\pi\)

C. \(2R^2\pi\)

D. \(8R^2\pi\)

**Question 3: **How will the area of the circle change if the radius is doubled?

A. Area is reduced by 2 times

B. Area reduced by 4 times

C. Area increased 2 times

D. Area increased 4 times

**Question 4: **Find the area of a circle whose circumference is \(10\pi (dm)\)

A. \(50\pi (dm^2)\)

B. \(25\pi (dm^2)\)

C. \(100\pi (dm^2)\)

D. \(64\pi (dm^2)\)

**Question 5: **A cow is kept in a square cage with a side of 5m. But because the barn is still for raising pigs, chickens, … so the owner had to tie the cow in a corner of the barn (the pole in the corner of the barn), the owner used a 1.5m-long rope to tie the cow. What is the area of the land where the cow can move?

A. \(25\pi (m^2)\)

B. \(3\pi (m^2)\)

C. \(\frac{9}{16}\pi (m^2)\)

D. \(4,5\pi (m^2)\)

## 4. Conclusion

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

- State the formula for calculating the area of a circle and the area of a circular fan.
- Apply learned formulas to solve some real-life problems.

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