2)Constructing Regular Polygons
3)Templates for Platonic Solid
The first lesson that I want to do involves the formula for finding the sum of the angles inside a polygon. Sum= (n-2)180 degrees. Where n = the number of sides of the polygon.
1) The angles of a triangle forms 180 degrees. We discovered this in a previous exploration involving triangles of different types.
2) In GSP , draw a four sided figure. Select the segment tool or place four points on you screen and connect the vertices to form the edges.
From one vertex, connect to all other vertices. (some may already be connected by a side)
How many triangles did you form?
Each triangle is 180 degrees. How many degrees do you have for all the triangles?
3) Draw a five sided figure.
From one vertex, connect to all other vertices.
How many triangles did you form?
Each triangle is 180 degrees. How many degrees do you have for all the triangles?
4) Draw an eight sided figure.
From one vertex, connect to all other vertices.
How many triangles did you form?
Each triangle is 180 degrees. How many degrees do you have for all the triangles?
5) Do you see a pattern in the number of sides and the number of triangles?
What is the equation you see for sides/triangles?
6) How can this pattern be used to find the sum of the angles in a polygon?
What is the equation formed using the above relationship between sides and triangles?
Constructing Regular Polygons
Now that you know the formula to finding the angle sum of any polygon, we want to use it to help us construct certain types of polygons. The type we are going to construct are called Regular Polygons. These are polygons with equal side lengths and angles.
This lesson is done on the Geometer's Sketchpad. This lesson will give the students some background work using the Rotation tool .
1) Construct a Regular Quadrilateral.
First you must find an angle measure.We learned that the sum of all the angles is (n-2)180.
A square has 4 sides, so (4-2)180 =______?
This is the measurement we will use to find one angle measure. How would you find the measure of one angle in a four angle figure?
Find the measure.
Now we need to go into GSP to construct the figure.
Select the segment tool on the tool bar.
Draw a segment.
Double click on one endpoint for the point of rotation.
Select the segment and the other endpoint.(make sure you hold down the shift key when selecting more than one object)
Under the heading Transform select Rotate...
Enter the angle amount we found in the above exercise. Hit enter.
This should give you a new segment the same length as the first one, and the angle measure needed to give you equal angles.
Repeat this process for the remaining sides.
Measure your sides. Are they the same length?
Measure your angle? Are they the same measure?
If the answer to the last two questions was yes, then you have just make a regular quadrilateral.
Constructing a Regular Quadrilateral
2) Construct a Regular Hexagon
As before, you must find the measure of one angle in a hexagon.
In GSP choose the segment tool.
Draw a segment.
Double click on one endpoint for the point of rotation.
Under the heading Transform, select Rotate...
Enter the measure of one angle in the hexagon. Hit enter.
Now repeat this process around the figure until you have a hexagon.
Constructing a Regular Hexagon
3) Construct another Regular Polygon of your choice. (example: Octagon)
4) Construct a Regular 50-gon.
What do you notice about the shape of the polygon as more sides are added?
Making templates for Platonic Solids
This lesson will be to make the templates used to make the 3-Dimensional objects called Platonic Solids. These solids were named in honor of the Greek philosopher Plato. There are only 5 Platonic Solids. They were discovered over two thousand years ago.
These lessons are done using The Geometer's Sketchpad. During these constructions, you will be using the Rotate and Reflect tools from the Transform heading. Remember that before you reflect or rotate, you must choose a line of reflection or point of rotation and have highlighted the objects you want to move. Students should have prior usage of rotation and reflection.
There are three basic shapes for the solids. The square, the triangle and the pentagon. All of these faces are regular polygons.
1) The Tetrahedron
This figure is made from four regular triangles used to form a triangular pyramid.
Construct an equilateral triangle. This will be the base of the tetrahedron.
There is an equilateral triangle off each edge of the base to form the sides of the pyramid.
Construct a template of the above type to print out.
Cut the template out, leaving the sides connected to the base. See if your template will make the tetrahedron.
If your template worked, explain why?
If not, what adjustments do you need to make concerning the sides or base?
Make the adjustments necessary to fix your template, and try again.
2) The Cube
The most common Platonic Solid is the cube. This figure has a base of a square, and all the other faces are squares of the same size.
Make a base of a square.
There are fours vertical faces coming off the base. These are of the same square shape.
There is also another base to form the top of the cube. This top needs to be connected to one of the vertical faces.
Construct a template for the cube.
Cut out your template to form the cube. Did your template work?
If so, explain why. If not, Describe changes you would need to make so that the template would form a cube.
Make the changes if necessary, and reprint the template.
3) Octahedron
This solid has its base in triangles. There are ____________ sides. We can tell this from the name of the polyhedron.
Construct an equilateral triangle.
Form the faces of the octahedron that are connected to the base or other sides.
Print and cut out the octhahedron. Does the template work?
If so, explain why it works.
If not, describe the changes needed to make the polyhedron work and do them.
4) Dodecahedron
This is the only platonic solid that uses the pentagon for it's base shape. This figure may need to be made from two of the same template.
Construct a regular pentagon.
Form pentagons off the base that will make half of the solid.
You can no either duplicate the template, or try to make the other half connected to the first half you just completed.
Print out and test your template. does it form a solid?
If so, explain why.
If not, describe the changes need to make your template work. Make the changes and try again.
5) Icosahedron
This figure has twenty faces of equilateral triangles.
Construct the base.
Form the other nineteen sides off the base to form the Icosahedron.
Print and cut out your solid. Does it work?
If so, explain why.
If not, describe the changes that need to be made.
Make the changes and try again.
Choose a spreadsheet program. My lesson is done using ClarisWorks.
1) Make a chart for Name,Vertices, Faces, and Edges.
2) Of the figures you made from the Platonic Solids lab, list their names on your chart.
3) Count the number of vertices on each figure. Put these numbers under the Vertices column.
4) Count the number of faces on each figure. Place these numbers under the Faces heading.
5) Count the number of edges and put the numbers in the appropriate column.
6) Try to find an equation that shows a relationship for the vertices, faces, and edges.
7) What equation did you came up with? Does this equation work for all of the Platonic Solids?
8) What about other types of space shapes?
Try constructing a square prism.
How many faces does the prism have? How many edges? Vertices?
Do these numbers fit into your equation?
9) What can you conclude about 3-Dimensional shapes in relation to their vertices, faces, and edges?
Example Spreadsheet for Euler's Formula
These solids are similar to Platonic Solids. The difference lies in the idea of using two or more different polygons for the faces. There are lots more solids of this type, but I just wanted to illustrate a few.
All of the following lessons will take more than one day in the lab, or divide the figures up among different students. All of the following labs are done on Geometer's Sketchpad. Students should have a good background in rotation and reflection on the computer
1) Cuboctahedron
This solid is formed from triangles and squares. At each vertex there is a triangle, square, triangle, and square in that order.
Start by making a square face.
From this we need to make two triangles and another square on the same vertex.
The end result should have the six faces of the cube plus the eight faces of the octahedron.
This will give us the fourteen faces for a cuboctahedron.
Does your template work?
If so, explain what you did to acheive the outcome.
If not, what adjustments do you need to make? Make these adjustments and try again.
2) Icosidodechedron
This solid is formed from triangles and pentagons. If we look back at the Platonic solids, which of these can we use to form this solid? (Hint: look at the names of the solids.)
Start by making a pentagon.
We should end up with all the faces for a dodecahedron and an icosahedron.
Does your template work for the Icosidodecahedron?
If so, how did you go about constructing you template?
If not, what adjustments need to be made?
Keep trying until your template gives you the desired figure.
3) Truncated Tetrahedron
This is a version of the Platonic Tetrahedron. This solid combines the tetrahedron and hexagons.
Form a hexagon.
Each vertx in the solid will have one triangle and two hexagons. From your starting hexagon, form the other sides.
Were you able to make a truncated tetrahedron the first time? If so, explain how you accomplished this. If not, describe what you did to make the necessary adjustments.
Making the Truncated Tetrahedron