LemonmathSelf-study sheet

Grade 9 · Lesson 5.3

Arc length. Area of a sector and an annulus

Name: Date: Time spent: min

Goals

Find the length of an arc of $n^\circ$ on a circle of radius $R$ .
Find the area of a sector and the area of an annulus.
Use the formulas in real-life problems (wheels, clocks, windscreen wipers).

Remember

Circumference $C=2\pi R$ ; area of a disc $\pi R^2$ .
Arc of $n^\circ$ : $l=\dfrac{\pi Rn}{180}$ .
Sector of $n^\circ$ : $S=\dfrac{\pi R^2n}{360}=\dfrac{lR}{2}$ .
Annulus: $S=\pi\left(R^2-r^2\right)$ .

Exercises

Exercise 1

Find the length of an arc of $45^\circ$ on a circle of radius $12$ cm (give your answer to 2 decimal places).

Exercise 2

Find the area of a sector of radius $6$ cm with an arc of $150^\circ$ (give your answer to 2 decimal places).

Exercise 3

Find the area of the annulus between two concentric circles of radii $5$ cm and $3$ cm (give your answer to 2 decimal places).

Exercise 4

An arc of $90^\circ$ has length $4\pi$ cm. Find the radius of the circle.

Exercise 5

Nam's bicycle wheel has diameter $70$ cm. How many metres does the bicycle travel when the wheel turns $100$ times (give your answer to 1 decimal place)?

Exercise 6

An annulus has area $40\pi$ cm², and the difference of its two radii is $2$ cm. Find the two radii. (Revise solving simultaneous linear equations.)

Exercise 7

(1.5 marks) A clock has a minute hand $10$ cm long and an hour hand $6$ cm long.

a) In $20$ minutes, how far does the tip of the minute hand travel (give your answer to 2 decimal places)?

b) Find the area of the sector swept by the minute hand in those $20$ minutes (give your answer to 2 decimal places).

c) In those $20$ minutes, through how many degrees does the hour hand turn, and how far does its tip travel (give your answer to 2 decimal places)?

Exercise 8 Challenge

Consider the circle $(O, 6\text{ cm})$ and a chord $AB=6$ cm. Find the area of the segment bounded by the chord $AB$ and the minor arc $AB$ (give your answer to 2 decimal places).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The length of an arc of

40^\circ

on a circle of radius

9

cm is

4\pi

cm.B.

2\pi

cm.C.

9\pi

cm.D.

\pi

cm.

Question 2. The area of a sector of radius

4

cm with an arc of

90^\circ

2\pi

cm².B.

16\pi

cm².C.

8\pi

cm².D.

4\pi

cm².

Question 3. The area of the annulus between two concentric circles of radii

7

cm and

4

cm is

33\pi

cm².B.

9\pi

cm².C.

65\pi

cm².D.

6\pi

cm².

Question 4. A wheel has diameter

60

cm. The circumference of the wheel is

30\pi

cm.B.

60\pi

cm.C.

120\pi

cm.D.

900\pi

cm.

Part II. True or false

Question 1. Consider a sector of radius

6

cm with an arc of

120^\circ

. Decide whether each statement is true or false.

a) The arc of the sector has length $4\pi$ cm.	T / F
b) The area of the sector is $12\pi$ cm².	T / F
c) The area of the sector equals the arc length times the radius.	T / F
d) The perimeter of the sector is $(4\pi+12)$ cm.	T / F

Part III. Short answer

Question 1. Find the length (in cm) of an arc of

36^\circ

on a circle of radius

10

cm (give your answer to 2 decimal places).

Question 2. Find the area (in cm²) of the annulus between two concentric circles of radii

6

cm and

4

cm (give your answer to the nearest whole number).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $l=\dfrac{\pi Rn}{180}$ .

Steps:

Substitute $R=12$ and $n=45$ .

Exercise 2.

What you need: $S=\dfrac{\pi R^2n}{360}$ .

Steps:

Substitute $R=6$ and $n=150$ .

Exercise 3.

What you need: $S=\pi\left(R^2-r^2\right)$ .

Steps:

Compute $R^2-r^2$ and multiply by $\pi$ .

Exercise 4.

What you need: $l=\dfrac{\pi Rn}{180}$ , so $R=\dfrac{180l}{\pi n}$ .

Steps:

Substitute $l=4\pi$ and $n=90$ .

Exercise 5.

What you need: in one turn, the bicycle travels the circumference $C=\pi d$ .

Steps:

Find the circumference.
Multiply by $100$ and convert to metres.

Exercise 6.

What you need: $R^2-r^2=(R-r)(R+r)$ .

Steps:

From the area, $(R-r)(R+r)=40$ , so $R+r=20$ .
Solve the simultaneous equations $R+r=20$ and $R-r=2$ .

Exercise 7.

What you need: the minute hand turns $6^\circ$ per minute and the hour hand $0.5^\circ$ per minute.

Steps:

Find the measure of the arc, then use $l=\dfrac{\pi Rn}{180}$ .
Use $S=\dfrac{\pi R^2n}{360}$ .
Do the same for the hour hand.

Exercise 8.

What you need: the area of the segment is the area of the sector $OAB$ minus the area of the triangle $OAB$ .

Steps:

Show that the triangle $OAB$ is equilateral, so the minor arc $AB$ measures $60^\circ$ .
Find the area of the sector and the area of the equilateral triangle.

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