Regularidades 3

	NOW FOR SOME GEOMETRY
	3rd year of secondary education

1. THE INTERIOR ANGLES IN A REGULAR POLYGON

In this window different regular polygons appear when you change the parameter "n". For practical reasons the highest value of "n" is 21. Look carefully at all the information given on the electronic board and then complete the activities that follow.

1.- Change parameter "n" and look at the different polygons that appear on the board. Then copy and complete the following table in your exercise book:

Polygon	nº of sides	nº of interior angles	interior angle	sum of angles	sum of angles/360
triangle	3	3	60º	180º	1/2
...	...	...	...	...	...
...	...	...	...	...	...

2.- Use the information in the table above to try and work out the following for an "n"-sided polygon:

a.- The sum of the interior angles

b.- The size of an interior angle, in degrees.

3.- Complete the following table:

polygon sides	interior angle	sum of interior angles
24
36
40
60
90

4.- Out of all the polygons that you have seen on the electronic board for which values of "n" was the interior angle a whole number? Can you explain why this is so?

2. DIAGONAL LINES IN A REGULAR POLYGON

Different types of regular polygons appear in this window, like in the window above. However, this time diagonal lines inside the polygons are indicated. Remember that a diagonal line is a segment which joins any two non-adjacent vertices in a polygon. In this case the highest value of "n" is fifteen as any values higher than this are too difficult to see clearly.

5.- Look carefully at the information given on the electronic board. Change the value of "n" and copy and complete the following table in your exercise book. (You decide on the number of rows):

Number of sides	nº of diagonals from each vertex	Total nº of diagonals
...	...	...

6.- How many diagonals are there from each vertex in an "n"-sided polygon? How many diagonals are there altogether?

7.- How many diagonals are there from each vertex and altogether in regular polygons with the following number of sides: 14, 25, 32 and 48?

3. A STAIRCASE OF SQUARES

In this window you can form a staircase with up to 8 steps, made by a series of squares. Change the "steps" parameter value and you will see how staircases of different heights appear.

8.- Use the arrows to change the "steps" parameter in the window. Draw a small table in your exercise book and write in the number of steps and the number of squares needed to form each staircase.

9.- Answer the following question in your exercise book: How many squares would staircases with the following number of steps be made up of: 10, 15 or 20? How many squares do we need to make a staircase with "n" steps?

10.- Imagine that the staircase goes up and then comes down. How many squares would be needed to make a staircase which is "n" steps high?

4. A PYRAMID OF SEGMENTS

In this window you can see how a triangular shape is formed using segments. These segments form small triangles which don't have a base. The "height" parameter is equivalent to the "steps" parameter in the exercise above.

11.- Change the "height" parameter and write down the number of segments which correspond with each "height" (or level) in your exercise book. Then answer this question: How many segments are needed to draw pyramids with the following number of levels: 8, 10 and 12?

12.- Try and work out an equation which allows us to calculate the number of segments needed in a pyramid with "n" levels. Check your answer by using the information you got in the exercises above. How many segments would be needed if we added the bases of the triangles as well?

	Josep Mª Navarro Canut

	ProyectoDescartes.org. Year 2013