**Narration**
1
00:00:01 --> 00:00:06
Hello. In this tutorial, we will work on **radians** and **sectors** using **Geogebra**.
2
00:00:07 --> 00:00:14
The objective of this tutorial is to introduce you to the Geogebra **Input Bar** and use of **commands** in the input bar through a lesson on radians.
3
00:00:15 --> 00:00:24
Geogebra beginners, please refer to **Introduction to Geogebra** and **Angles and Triangles Basics** on the spoken-tutorial.org web site.
4
00:00:25 --> 00:00:34
In this tutorial, I have worked on **Ubuntu version 10.04 LTS** and **Geogebra version 3.2.40.**
5
00:00:35 --> 00:00:38
In this lesson: we will understand what a **radian** means and how to draw a **radian**,
6
00:00:39 --> 00:00:43
understand the relationship between length of an **arc** and the angle it subtends
7
00:00:44 --> 00:00:48
and complete an assignment to calculate the area of a sector.
8
00:00:49 --> 00:00:59
We will use the following tools in Geogebra- **Circle with Center and Radius**, **Circular Arc with Centre between Two Points** and **Segment between Two Points**.
9
00:01:00 --> 00:01:10
The drawing commands can be used in another way by typing commands in the **Input bar** as well.
10
00:01:11 --> 00:01:17
In this Geogebra window, now we will draw a circle of radius 5 units using the **Circle with Centre and Radius**.
11
00:01:18 --> 00:01:27
I click on **Circle with Center and Radius**, we choose the centre to be at origin, radius 5 units.
12
00:01:28 --> 00:01:35
I will now plot two points **B** and **C** on the circle.
13
00:01:36 --> 00:01:46
Now we will complete an arc between these two points, I click on the **Circular Arc with Centre between Two Points** to draw an arc.
14
00:01:47 --> 00:01:59
I click on **A** the centre, **B** and **C**. This completes the arc. Notice that the arc length is d=5.83 units.
15
00:02:00 --> 00:02:09
Now we will delete this arc and construct it another way. The arc can also be constructed by entering a command in the **Input** bar here.
16
00:02:10 --> 00:02:29
This rectangular box, here, is the **Input** bar. There are 3 drop down boxes next to the input bar. Here you can introduce some functions, define some parameters and this is the **command** key in which you can complete drawings in the Geogebra window here.
17
00:02:30 --> 00:02:40
Now I will start typing **arc** here, you will notice that it completed the command for me. I can also look up this command from the drop down box here.
18
00:02:41 --> 00:02:56
I click on **arc**, you will notice that the command appears here, with square brackets. If I click in the middle of the square brackets and hit **Enter**, the syntax for this command will appear here.
19
00:02:57 --> 00:03:03
The syntax that we will use for '*arc* now is to define the circle and the two points.
20
00:03:04 --> 00:03:09
We will have to define the name of the circle and the two points between which we want the **arc**.
21
00:03:10 --> 00:03:23
From the **Algebra View** we can see that the circle is referred to as lower case **c** and the points between which we want to draw the **arc (B,C)** both in upper case.
22
00:03:24 --> 00:03:36
So we will type the command here **Arc[c,B,C]** and hit Enter. Geogebra is **case sensitive**.
23
00:03:37 --> 00:03:45
Now lets change the color and thickness of the arc that we have joined, from object properties here.
24
00:03:46 --> 00:04:04
We will go to Color, we define it as red. From Style, we increase the thickness.
25
00:04:05 --> 00:04:10
Notice that the arc is now appearing as a bold, red thick arc.
26
00:04:11 --> 00:04:16
Now we will draw two line segments AB and AC. We will again do this in two ways.
27
00:04:17 --> 00:04:27
We click on the **Segments between Two Points** tool here and click on **A** and **B**. This completes the segment **AB**.
28
00:04:28 --> 00:04:39
We can also enter a command for the segment from the input bar. we will do **Segment[A,C]** to complete the segment **AC**.
29
00:04:40 --> 00:04:46
Now we have completed the **arc BC**, drawn segments **AB** and **AC** and the sector **BAC**.
30
00:04:47 --> 00:04:57
We will now define the angle subtended at **A** by **arc BC**. we will call this angle **α**. We will choose it from the drop down box here.
31
00:04:58 --> 00:05:09
Angle command is **angle[B,A,C]**.
32
00:05:10 --> 00:05:17
We will follow the standard angle naming convention when we define angles in Geogebra as well.
33
00:05:18 --> 00:05:29
We notice that the value of **α** here, subtended at the center, is 66.78 degrees.
34
00:05:30 --> 00:05:39
Now, "one radian is defined to be the angle subtended at the center when the length of the arc subtending the angle is equal to the radius of the circle".
35
00:05:40 --> 00:05:48
If we define the angle unit to be in radians, by going to the **Options** here and defining **Angle Units** to be **Radians**.
36
00:05:49 --> 00:06:03
We will find that the value of α now is 1.17 rad. We will now change the **arc** length to bring it closer to 1 rad.
37
00:06:04 --> 00:06:16
Notice that the **arc** length is d=5 units and the value of **α**, the angle subtended at the center, is 1 rad.
38
00:06:17 --> 00:06:28
We defined 1 rad, we also saw that this is the angle that will be subtended when the arc length is equal to the radius.
39
00:06:29 --> 00:06:40
What is the value of 1 rad in degrees? I just zoomed it out a little bit.
40
00:06:41 --> 00:06:52
Now let's change the length of this arc to that of a semi circle, so the arc length is [π a] where **a** is the radius of the circle.
41
00:06:53 --> 00:07:02
Before that, I will redefine the angle unit to be **Degrees** because we want to find the value of 1 rad. in degrees.
42
00:07:03 --> 00:07:12
We notice that when the **arc** length is [π a] that is a semi circle, the value of α is 180.21 degrees.
43
00:07:13 --> 00:07:26
And, if I complete this circle we notice that the angle of α will be almost 360 degrees.
44
00:07:27 --> 00:07:34
So, we notice from these two that the value of 1 rad will be 57.32 degrees.
45
00:07:35 --> 00:08:02
Now we will understand the relation between **arc** length, radius and the angle subtended. For that, we will define another angle value **θ** in radian by dividing the value of α/57.32.
46
00:08:03 --> 00:08:14
Notice that the value of **θ** is actually the value of the angle in radians. However, it appears with a degree symbol here because of a formatting difficulty.
47
00:08:15 --> 00:08:28
We will continue to use **θ** like this and not change the angle unit as radians, because we want to illustrate a formula using the arc length and angle subtended.
48
00:08:29 --> 00:08:35
Due to a formatting difficulty this formula can be explained only in this way.
49
00:08:36 --> 00:08:51
Now we will insert text in the Geogebra window to introduce the formula that relates the arc length to the angle subtended.
50
00:08:52 --> 00:09:33
For an introduction on how to write text, please refer to the tutorial **Angles and Triangles Basics**.
51
00:09:34 --> 00:09:57
Now, notice that when I change the arc length, you will notice that the value of **θ** changes and the relation between arc length and the angle subtended goes as d=r.θ where 'd' is the arc length, 'r' is the radius of the circle and 'θ' is the angle subtended at the center, in radians.
52
00:09:58 --> 00:10:09
Now we will look at an assignment to reinforce our understanding of what we have learned.
53
00:10:10 --> 00:10:17
Using what we have learned, show how the area of a sector will be **Area = ½ a^2 θ**
54
00:10:18 --> 00:10:29
where **a** is the radius,**θ** is the angle subtended at the center in radians and the formula is **Area** = ½ **a^2** **θ**.
55
00:10:30 --> 00:10:39
A small hint to complete this assignment is to compare the area of the sector to the quadrant.
56
00:10:40 --> 00:10:54
The assignment when drawn, will look like this. We want to calculate the area of the sector here by comparing it with the quadrant here.
57
00:10:55 --> 00:11:05
I would like to acknowledge the spoken tutorial project which is a part of talk to a teacher project, supported by the National Mission on Education through ICT, MHRD, Government of India.
58
00:11:06 --> 00:11:11
More information can be found here. Thank you for joining me on this tutorial of Geogebra. This is Ranjani Ranganathan.