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

Since this tutorial is very long I have split it into two parts. In Part I, we will inscribe a regular polygon in a circle, increase its number of sides, and investigate the relationship between their areas. In Part II (to be posted next week), we will use the polygon circumscribing the same circle and 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. We will need the Algebra window and the Axes so be sure that they are displayed. If not, use the View menu from the menu bar to show them.

2. We will only need to label of the points, so we will not GeoGebra automatically label other objects. To do this, click the Options menu, click Labeling, then click All New Points Only.
3. To create a slider r for the radius of our circle, select the Slider tool, then click anywhere on the drawing pad 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.

Figure 1 – The Slider dialog box.

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 our inscribed polygon.

6. Next, we do not want any new objects to have labels. To do this, click the Options menu, click Labeling then click No New Objects.

7. To construct a circle with center A and radius r, type circle[A,r] . Move slider r and see what happens.

8. To construct the intersection of the circle and the x-axis, type (r,0) in the input box and press the ENTER key.

9. Now, we compute for the central angle of our 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 our 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 degree sign by selecting it at the drop down list box right next to the input box.
10. To create angle BAB’, click the Angle with Given Size tool, click point B and click 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 and choose the counter clockwise button and then click the OK button. 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. This means, that we want to inscribe a polygon with n sides, then click the Ok button.. Now, drag slider n and see what happens. If you set n to 30, your drawing should look like the one shown in Figure 3.

Figure 3 - A circle with an inscribed 30-sided polygon.

15. Our 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 Properties from the context menu.

16. In the Properties 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 and uncheck the Show Label check box to hide the labels of all the sides of the polygons.

Figure 4 - The Properties dialog box.

17. Now, using the text tool, we will display the area of the circle and the area of the inscribed triangle. To display the area of the circle, click the Insert Text tool and click anywhere and type the following:

The area of the circle with radius ” + r + “is ” + pi*r^2.

The blue text enclosed by double quotes are constants and will exactly appear as they are. The red texts r and pi*r^2 are variables and will display numbers based on the value of the slider r and the result of the computation. GeoGebra interprets pi as the mathematical constant which approximately equals 3.1416. Note that constants are always enclosed by double quotes. Constants and variables are always separated by the + symbol.

18. Use the text tool to display the entire polygon. In the Text tool, type

The area of the inscribed polygon is “ + poly1

Note that poly1 is the area of our polygon (see the Algebra window). Adjust the positions of the text as needed. Move the sliders and observe what happens.

19. Your drawing should look like the Figure below. (The font of the text has been resized to make it more visible in the drawing).

Figure 5 - Final output.

20. What is the relationship between the area of the circle and the and the area of the inscribed polygon?

You may also want to view other GeoGebra tutorials here: GeoGebra Tutorial Series.

Last modified: Tuesday, 7 December 2010, 05:15 PM
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