I don't think I know of any other project that has more Geometry in it than this one. Students begin with a large circle, and as they follow the directions to fold the paper in different ways, they will find an equilateral triangle. Then, folding on the midpoints of the sides of the triangle, they will create 4 smaller equilateral triangles, or one triangle and an isosceles trapezoid. In the next step of the project, they will find a tetrahedron, followed by a hexagon, culminating in a truncated tetrahedron, experiencing and learning Geometry all the way along!

This can be a great "quickie" project for the day before Christmas vacation. If you give the students red and/or green paper with which to do this project, they can use the project as a package to enclose a small Christmas gift, candy or trinket.

To enrich the project, if time permits, students can answer all of the geometric questions below (in bold print along the way). What a great way to review the concepts of perimeter, surface area, and volume!

1. Start with a circle, cut out of stiff paper that has about a three-inch radius. Fold to find the center: Fold two diameters; where they intersect is the center of the circle. (Why does this method work? If given the radius = 12, then find the area and the circumference of the circle.)

2. Refer to the diagrams below in doing the three steps below. Fold a random point of the circle to the center of the circle and crease the folded edge. Make another fold to the center of the circle so that the folded edge forms an angle with the vertex on the circle. Make a third fold to the center so that the folded edge intersects the other two edges on points on the circle. You now have an equilateral triangle. (Why is this an equilateral triangle? Find the length of an altitude. Find the perimeter and area)

3. Fold each vertex to another vertex, one at a time, to find the midpoints of the three sides of the triangle. Fold one vertex to the midpoint of the opposite side to form an isosceles trapezoid. (What are the properties of equilateral triangles? Why is this an isosceles trapezoid? Find the perimeter and the area of the large triangle; find the area of one of the smaller triangles; discuss the properties of isosceles trapezoids; find area ratio of trapezoid to the triangle; find the area using the formula for the area of the trapezoid, then find area by adding the areas of the 3 triangles, to verify the area you found for the area of the trapezoid; find the perimeter of the trapezoid.)

 

4. Notice that the trapezoid is formed by three triangles. Fold one of the triangles onto the middle triangle and you have a rhombus. Fold again to get an equilateral triangle. (Find the perimeter and area of each figure shown below.)

 

5. Let the three triangles open up and come to a point - it's now a regular triangular pyramid or tetrahedron. (Why does this form a tetrahedron? Find the surface area of the pyramid by using the formula; then check that answer by finding the area of one face and multiplying by 4. Explain why you get the same answer with either method. Find the volume of the pyramid)

 

6. Open the paper back out to the large equilateral triangle made in step 3. Fold each vertex to the center and you have a regular hexagon. (What is the definition of a regular hexagon? Find the area of the hexagon.)

 

7. Raise the vertices a little and gently push toward the center.If this fails try resting the paper in the palm of your hand and tuck the three vertices together. Either way it will form a truncated tetrahedron; also called a frustom ("the part of a conical solid left after cutting off a top portion with a plane parallel to the base" from http://dictionary.reference.com/.).(Find the lateral and total surface areas of the frustom. Find the volume of the frustum.)

(If you glue 20 of these together, you will have an icosahedron. If you would like to see what an icosahedron is, click on the following link: http://mathworld.wolfram.com/Icosahedron.html

8) Unfold your truncated tetrahedron and this is what you will see. There is a lot of geometry here! (Find and label all segments with their lengths and all angles with their degree measure.)

 

This project can be made into a worksheet by assigning a numerical value to the radius of the original circle, and then asking the students to calculate the lengths of the sides at each step, the measures of the angles, the area and/or volume of each solid formed along the way.

Answers: Given the radius of the circle is 12 inches then:

Step 1) circumference of circle = 24 pi inches., area of circle = 144 pi square inches

Step 2) altitude of equilateral triangle = 18 inches, perimeter of equilateral triangle=36 times the square root of 3 inches, area of the equilateral triangle = 108 times the square root of 3 square inches

Step 3) perimeter of the trapezoid = 30 times the square root of 3 inches, area of the trapezoid = 81 times the square root of 3 square inches, area ratio triangle ABC:trapezoid BDEC=3:4

Step 4) Area of the rhombus=54 times the square root of 3 square inches

Step 5) Lateral Surface Area = 81 times the square root of 3 square inches, total surface area=108 times the square root of 3 square inches, volume=54 times the square root of 6 cubic inches

Step 6) Perimeter=12 times the square root of 6 inches, area=36 times the square root of 3 square inches