## GeoGebra Tutorial 14 – Sliders and Circle-Area Approximation

This is the 14th tutorial in the GeoGebra Tutorial Series. If this is your first time to use GeoGebra, I suggest that you read the GeoGebra Essentials series first.

In this tutorial, we use GeoGebra to approximate the area of a circle.  This strategy of approximating the area of a circle was used by the Greek mathematician Archimedes.

Since this tutorial is long, we split it into two parts. In Part I, we inscribe a regular polygon in a circle, increase its number of sides, and investigate the relationship between the areas. In Part II we will circumscribe the circle with a regular polygon increase its number of sides to approximate the circle’s area.  Before following the tutorial step-by-step, click here to view the final output.

Part I – Creating an Inscribed Polygon

Step-by-Step Instructions

 1. Open GeoGebra click on Algebra & Graphics in the Perspectives menu. 2. In this tutorial, we want all new objects to be labeled. To do this, click the Options menu, click Labeling, then click on All New Objects. 3. Next, we create  slider r that will determine the radius of the circle that we are about to create.  Select the Slider tool, then click anywhere on the Graphics view to display the Slider dialog box. 4. In the Slider dialog box, type r in the Name box, type 0.1 in the min box, and leave the max value as 5 and increment as 0.1, then click the Apply button. 5. Create another slider name it n, set the minimum to 3, maximum to 30 and increment to 1. Slider n will determine the number of sides of the inscribed polygon. 6. Next, we create point A on the origin. To do this, select the Intersect Two Objects Tool, click the x-axis and then click the y-axis. 7. To construct a circle with center A and radius r, type circle[A,r] . Move slider r and observe what happens. 8. To construct point B which is the intersection of the circle and the x-axis, type B = (r,0) in the input box and press the ENTER/RETURN key on the keyboard. 9. Now, we compute for the central angle of the inscribed polygon. To do this, we divide 360 by n. For example, if we want to have an equilateral triangle, we must divide 360 by 3, which will be the central angle. To do this, type a = (360/n)° then press the ENTER key. The degree sign, tells GeoGebra that a is an angle measure. You can display the ° sign can be written by clicking the $\alpha$ button at the right of the input box. 10.  To create angle BAB’, click the Angle with Given Size tool, click on point B and then click on point A. This will display the Angle with Given Size dialog box. 11.  In the Angle dialog box, type a in the Angle text box,  choose the counterclockwise option, and then click the OK button. If you set r to 3, your drawing should look like the one shown in Figure 2. Figure 2 – Central angle BAB’. 12.  To hide the angle measure (green sector), right click it then click Show Object. 13.  To construct the inscribed polygon, select the Regular Polygon tool, click B and then click B’. This will display the Regular Polygon dialog box. 14.  In the Regular Polygon dialog box, type n and then click the OK button. Now, drag slider n and see what happens. If you set n to 30 and a to 3, the figure should look like Figure 3. Figure 3 – A circle with an inscribed 30-sided polygon. 15.  The problem now is to hide the labels of all the points and the segments. With n set to 30, right click the polygon, then click Object Properties from the context menu to display the Preferences dialog box. 16.  In the Preferences dialog box, select the Basic tab, click Point (be sure that the Point text is highlighted) in the Objects list, and uncheck the Show Label check box. This will hide the labels of all the points. Now, click Segment text in the Objects check box and uncheck the Show Label check box to hide the labels of all the sides of the polygons, then click the Close button. Figure 4 – The GeoGebra Preferences dialog box.

Exercise:

1. Move the slider and observe what happens.
2. Using the text tool, display the area of the circle and the area of the inscribed triangle.Your drawing should look like Figure 5.

Figure 5 – Final Output

## GeoGebra Tutorial 11 – Sliders and Graphs of Trigonometric Functions

This is the eleventh tutorial in the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you may want to read the GeoGebra Essentials Series.

In this tutorial (now updated to GeoGebra 4.2), we use sliders to explore the effects of the parameters a, b, c and d of the graph f(x) = a sin(bxc) + d, g(x) = a cos(bxc) + d and h(x) = a tan(bxc) + d. We also learn about a new tool, the Checkbox to Show/Hide Objects tool. The output of this tutorial is shown in Figure 1.  If you want to explore first before following the tutorial, the GeoGebra applet can be viewed here.

Figure 1 – The graphs of the sine, cosine and tangent functions with similar values of a, b, c and d.

Although this tutorial is the eleventh of the GeoGebra Tutorial Series, it is a stand-alone tutorial. You may follow it step-by-step without having to learn the previous ten.

Instructions

 1.) Open GeoGebra and click on the Algebra & Graphics at the Perspectives menu. 2.) First, we create assigned values to a, b, c and d. These numbers will be our slider later. To assign a number to a, type a = 1 in the Input bar, then press the ENTER key. Now, construct numbers b, c, and d by typing b = 1, c = 1, and d = 1 and press the ENTER key after each equation. 3.) To create sliders a, b, c and d, right click each one of them in the Algebra view and click Show object from the context menu. 4.) Before graphing the functions, change the interval of the x-axis from 1 to π/2. To do this, right-click any blank space on the Graphics view and click on Graphics… from the context menu to display the Settings dialog box. 5.) In the Settings dialog box, select the Graphics section,  click the x-axis tab, check the Distance check box, select π/2 from the Distance drop-down list box, and then close the dialog box. 6.) Now, to graph the sine function, type f(x) = a*sin(b*x-c)+d in the Input bar, then press the ENTER key. In GeoGebra, the * symbol stands for multiplication. 7.) To graph the cosine and tangent function, type g(x) = a*cos(b*x-c)+d,then press the ENTER key and type h(x) = a*tan(b*x-c)+d then press the ENTER key. 8.) Move sliders a, b, c and d. What do you observe? How does a, b, c and d affect the graph of the sine function, the cosine function and the tangent function? 9.) Next, we create three check boxes that will show or hide the graphs of the three trigonometric functions. To create a check box that will show/hide the sine function, select the Check Box to Show/Hide Object tool, then click anywhere on the Graphics view to display the Check Box to Show/Hide Objects dialog box. 10.)  In the Check box dialog box, type Show/Hide Sine Function in the Caption box,  click the sine graph in the Graphics view, and then click the  Apply button to finish. Notice that the sine function appears on the list box. Figure 5 – The Check Box to Show/Hide Objects dialog box. 11. )   Select the Move tool and click the check box several times. What do you observe? 12. )   Using steps 9-10, create two more check boxes for the cosine function and the tangent function. 13.)    Next, we show the grid. We can use the grid to approximate the x and y values of a particular point. To do this, click the View menu, then click Grid. 14.)   Lastly,  change the color of the graphs to distinguish them easily.

## GeoGebra Tutorial 7 – Sliders and Rotation

This is the seventh tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, please read the GeoGebra Essentials Series.

In the Graphs and Sliders posts (click here and here ), we have discussed how to use number sliders.  In this tutorial, we use the Angle slider to rotate a triangle in order to show that its angle sum is 180 degrees. This is the same GeoGebra worksheet shown in my Parallel Lines and Transversals post, but we will change some of the labels. Although this tutorial is the seventh of the GeoGebra Tutorial Series.

Figure 1 – Rotated triangles using sliders.

Construction Overview

The construction will start by drawing line AB and constructing triangle ABC using the Polygon tool. Afterwards, we reveal the interior angle measures of the triangle and create two angle sliders namely $\alpha$  and $\beta$. Next, we rotate the triangle 180 clockwise about the midpoint of BC producing triangle A’B’C’ (see Figure 1-B). We then repeat the process, and rotate triangle A’B’C’ 180 degrees clockwise about the midpoint of A’C’ to produce A’’B’’C’’.

Part I – Constructing Triangle ABC

 1.) Open GeoGebra and select Geometry from the Perspectives menu. 2.) Click the Line through Two Points tool, and click two distinct locations on the Graphics view to construct line AB. 3.) If the labels of the points are not displayed, click the Move tool, right click each point and click Show label from the context menu. 4.) Click the New Point tool and construct a point C not on line AB. 5.) Display the name of the third point. GeoGebra would automatically name it C, otherwise right click and rename it C. 6.) Click the Polygon tool and click the points in the following order: point A, point B, point C, and click again on point A to close the polygon. Your drawing should look like Figure 1. Figure 2 – Triangle ABC on line AB. 7.) Move the vertices of the polygon. What do you observe? 8.) Now we construct two angle sliders $\alpha$ and $\beta$. To do this, click the Slider tool, and click on the Graphics view. 9.) In the Slider dialog box (see Figure 3), choose the Angle radio button, and then leave the name angle as $\alpha$.  In the Interval tab, choose 0° as minimum, 180° as maximum and 1°, and then click the Apply button when finished. Figure 3 – The Slider dialog box 10.) Using steps 8-9, create another slider with the same specifications shown in Figure 3 and name it $\beta$. You can find the Greek letters by pressing the $\alpha$ button located at the right of the text box. 11. ) We reveal the angle measures of the interior angles of the triangle, the change the colors of the angle symbols (green sectors). To do this, click the Angle tool and then click the interior of triangle ABC. 12.)  We now hide the measures of the angles. To do this, right click each angle symbol and uncheck Show label from the context menu. 13.  We set angle colors: angle A red, angle B blue and angle C green. To change the color of the angle symbol of angle A, right click the angle symbol (not point A) and click Object Properties from the context menu to display the Preferences window. 14.  In the Preferences window, click the Color tab and choose the color you want from the color palette then click the Close button. 15.  Change the color of angle B to blue and leave angle C as is.  Your drawing should look line Figure 1-A after step 15.

Part II – Rotating the Triangle

We already have the sliders ready. The next thing that we will do is to rotate the triangle. The idea is to create a rotation point. Our choice would be the midpoint of BC. That is because if we rotate ABC by 180 degrees producing A’B’C’, angle A’C’B’ will be adjacent to angle ABC (see Figure 1-B). This is also the idea when we rotate A’B’C producing A’’B’’C’’.

.

 1.) To construct D, the midpoint of BC, click the Midpoint or Center tool, and click side BC (the segment, not the points). 2.) Note that we want ABC to rotate around D $\alpha$ degrees clockwise. To do this, choose Rotate around a Point by Angle tool, click the interior of the triangle and click point D to reveal the Rotate Object dialog box. 3.) In the Rotate Object around Point by Angle dialog box, change the measure of the angle to $\alpha$, choose the clockwise radio button, and then click the OK button. Figure 4 – The Rotate Dialog Box 4.) Now move slider $\alpha$. What do you observe? 5.) Adjust slider $\alpha$ to 90 degrees, and show the labels of the vertices of the rotated triangle. (Refer to Part I – Step 3). 6.) While the triangle is still rotated 90 degrees, click the Angle tool and click the interior of triangle A’B’C’. Hide the labels of the angles symbols. 7.) Change the colors of the angle measures. Refer to Part I – Steps 13 through 15. Be sure that angle A and A’ have the same color, B and B’ have the same color, and C and C’ have the same color. Your drawing should look like the drawing in Figure 6. Figure 5 – The Rotated Triangle

Part III – Creating the Third Triangle

The idea of creating the third triangle is basically the same as that of creating the second triangle, so I will just enumerate the steps and left the construction as an exercise.

1. Get the midpoint of A’C’. (Refer to Part II Step 1)
2. Rotate triangle A’B’C’ $\beta$ degrees clockwise around the midpoint of A’C’.  (Refer to Part II – Steps 2 – 3).
3. Reveal the labels of the vertices of the third triangle which is A’’B’’C’’.  (Refer to Part II – Step 5 and Part I – Step 3).
4. Reveal the angle symbols of triangle A’’B’’C’’.  (Refer to Part I – Step 11)
5. Hide the labels of the angle symbols, and change the colors of the angle symbols of triangle A’’B’’C’’. (Refer to Part I – Step 13-15)

The explanation of the theory behind this construction is in my Parallel Lines and Transversal post.