## Math 9 Chapter 3 Lesson 9: Length of circles, arcs

## 1. Summary of theory

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

The length \(C\) of a circle with radius \(R\) is calculated by the formula: \(C = 2\pi R\)

If \(d\) is the diameter of the circle \((d=2R)\) then \(C = πd\)

### 1.2. Formula for calculating arc length

On a circle of radius \(R\), the length \(l\) of an arc with measure \(n^0\) is calculated by the formula: \( l\)= \(\dfrac{\pi Rn}{180}\).

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **

Fill in the blanks with appropriate expressions (…) in the following series of arguments:

Circle of radius R (corresponding to arc 360^{o}) has a length of…

So arc 1^{o}, radius R has length \(\displaystyle {{2\pi R} \over {360}} = …\)

Derive the supply n^{o}, the radius R has a length of…

**Solution guide**

Circle of radius \(R\) (corresponding to arc 360^{o}) has a length of \(2\pi R\)

So arc 1^{o}, radius R has length \(\displaystyle {{2\pi R} \over {360}} = {{\pi R} \over {180}}\)

Derive the supply n^{o}, radius R has length \(\displaystyle {{\pi Rn} \over {180}}\)

**Verse 2: **Given a circle (O), radius R = 4cm. Calculate the circumference of a circle?

**Solution guide**

The circumference of the circle is:

C = 2πR = 2π.4 = 8π (cm)

### 2.2. Advanced exercises

**Question 1:** The plow has two rear wheels that are larger than the front two. Know that when inflated, the front wheel has a diameter of 0.8m, the rear wheel has a diameter of 1.5m. Ask the rear wheel to roll 16 times, how many times can the front wheel roll?

**Solution guide**

A wheel that has rolled one revolution means that it has traveled a length that is the circumference of the wheel.

The circumference of the front wheel is \(C_1=\pi d=0.8 \pi (m)\)

The circumference of the rear wheel is \(C_2=\pi d=1.5 \pi (m)\)

The rear wheel has rolled 16 times, meaning it has gone: \(s=1.5 \pi.16=24\pi (m)\)

Then the front wheel will roll the number of revolutions: \(\frac{24\pi}{0.8\pi}=30\)(round)

**Verse 2:**

a) Calculate the arc length \(60^0\) of a circle with radius \(2 dm.\)

b) Calculate the circumference of a bicycle rim with diameter \(650mm.\)

**Solution guide**

a) Change \(R = 2dm = 20cm\).

The arc length \(60^\circ \) is \(l = \dfrac{{\pi Rn}}{{180}} = \dfrac{{\pi .20.60}}{{180}} = \dfrac{ {20\pi }}{3}\) (cm)

b) The diameter \(d = 650mm = 65cm\) so the circumference of the bicycle rim is \(C = \pi d = \pi .65 = 65\pi \,\left( {cm} \right)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given two circles with radii \(R = 1km\) and \(r = 1m.\) If the length of each circle increases by \(1m\), the radius of each circle increases by \(1m\) respectively. how much extra? Please explain.

**Verse 2: **Calculate the length of the circumcircle:

\(a)\) A regular hexagon with sides \(4cm;\)

\(b)\) A square whose side is \(4cm;\)

\(c)\) An equilateral triangle with side \(6cm.\)

**Question 3: **The equator is a great circle of the Earth with a length of about \(40 000km.\) Calculate the radius of the Earth.

**Question 4:** Mat – xco – and has a latitude of \(56^\circ\) North. Find the length of the arc of meridians from Moscow to the Equator, knowing that each meridian is a half of the great circle of the Earth, about \(20 000km.\) in length.

### 3.2. Quiz exercises

**Question 1:** The length of a semicircle of diameter 8R is

A. \(\pi R\)

B. \(2\pi R\)

C. \(4\pi R\)

D. \(8\pi R\)

**Verse 2:** Agricultural tractors have two larger rear wheels than the front two. When inflated, the rear wheel has a radius of 0.75m, the front wheel has a radius of 0.5m. If the tractor has traveled 471m, how many turns will the rear wheel and front wheel roll? (Know \(\pi\approx 3.14\))

A. 100 turns and 150 turns

B. 120 laps and 140 laps

C. 100 turns and 120 turns

D. 120 turns and 150 turns

**Question 3: **Arc Length 30^{0} of a circle with a diameter of 10m is:

A. \(\frac{5}{6} m^2\)

B. \(\frac{5\pi}{6} m\)

C. \(\frac{5\pi}{6} cm\)

D. \(\frac{5\pi}{3} m\)

**Question 4:** Radius of a circle with arc length 30^{0} is that \(2\pi\) is:

A. \(12\)

B. \(18\)

C. \(10\)

D. \(15\)

**Question 5:** A circle with center O has perimeter \(18\pi\), arc AB on the circle is \(6\pi\) .Calculate angle \(\widehat{AOB}\)

A. \(\widehat{AOB}=90^0\)

B. \(\widehat{AOB}=150^0\)

C. \(\widehat{AOB}=60^0\)

D. \(\widehat{AOB}=120^0\)

## 4. Conclusion

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

- Remember the formula for calculating the length of a circle.
- Know how to calculate the length of an arc.
- Know how to apply formulas to calculate unknown quantities in formulas and solve some real-life problems.

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