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Title of the tutorial: Radian Measure

Author: Ranjani Ranganathan

Key words: Length of an arc, radians and sectors, Circle with centre and radius, Circular arc with centre between two points, Segment between two points, Input Bar, video tutorial.

Visual Cue Narration
Slide 1 Hello. In this tutorial we will work on radians and sectors using Geogebra.

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.

Geogebra beginners , please refer to the following tutorials

Introductory and Basic tutorials

Slide 2 Our lesson will cover the following:

Understand the relationship between length of an arc and the angle it subtends

Complete an assignment to calculate the area of a sector

Slide 3 We will use the following tools in Geogebra

Drawing tools

- Circle with centre and radius

- Circular arc with centre between two points

- Segment between two points

Input Bar

The above drawing tools can be used in another way by typing commands in the input bar

Geogebra window - construction Now we will draw a circle of radius 5 units using the circle with centre and radius

We have a circle “c” centered at A.

I will now plot two points B and C on the circle.

I want to draw an arc between these two points.

I click on the “Circular arc with centre between two points” and draw an arc.

I click on B, A (the centre) and C and here is my arc.

You will notice that the arc length is given by d = 4.58 units.

Now we will remove this arc and do it another way.

I can also draw an arc by using the input bar below.

This is the input bar.

I will type the command here for drawing an arc.

I can start typing arc..and it will complete the command for me.

Or I can go the command prompt and look up the arc.

I click on Arc; it will appear in the input box with square brackets.

If I click in the middle of the square brackets, the syntax will appear.

We have to define here the name of the circle “c”, and the two points between which I want the arc.

From the algebra view we can see that the circle is called lowercase “c”. Similarly the points on the circumference of the circle are uppercase B and C, Geogebra is case sensitive..

The command will be Arc[c,B,C]. The arc length is units

Let us change the colour of the Arc in the object properties.

Now we will draw line segments AB and AC.

We will again do it in two ways.

We click on the segment tool and complete the segment like this.

We can also enter it in the input bar.

Now we have completed the arc BC and the sector BAC.

We will now define the angle subtended at A by arc BC.

Notice that we can also introduce parameters and functions in these drop down boxes next to the Command.

We will introduce “α” for the angle measure from this dropdown box.

We will use the input bar to define it like this angle[B,A,C] – points must be written in clock-wise order. The syntax is similar to the angle naming convention we use.

We notice that the value of α is ---- ˚.

One radian is defined as the angle subtended at the centre when the length of the arc subtending the angle is equal to the radius of the circle.

If we define the angle unit to be in radians, by going to the Options in the menu bar and selecting Angle Unit as radians, we will find that the value of α is...

We will change the length of the arc to get the angle to be 1 radian.

Screenshot of 1 radian construction When constructed, one radian will look like this
Geogebra window - construction We defined one radian. We also saw that this is equal to 57.32°.

How is this so?

If I change the arc length to semi-circle, arc length will be πr. Therefore, angle subtended is π rad = 180°

If I change the arc length to the full circle, arc length will be 2πr. Therefore, angle subtended is 2π rad = 180°

Therefore, value of one rad = 57.3°

Geogebra insert text Now we will understand the relation between arc length, radius and angle subtended

Arc length, “d”, radius “a” and angle subtended in radian “θ” is given by

d = a x θ

We will insert a text in the geogebra window to introduce this animation.

For an introduction on how to write text please refer to the tutorial “Angles and Triangles Basics”.

Now we will notice here that the arc length varies according to the angle subtended in radians.

Slide 4 Now we will look at an assignment

Using what we have learnt, show how the area of a sector will be

Area = ½ “a2” “θ”

where “a” is the radius and “θ” is the angle subtended in radian.

One hint for solving this assignment is to compare the area of this sector to the area of the quadrant.

Screenshot of assignment The geogebra drawing of the assignment looks like this
Slide 6 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