Key Question
What are the geometric
properties of a regular
tetrahedron, and how do
these compare to a bird
tetrahedron?
Math
Properties of polyhedra
faces
edges
vertices
Surface area
Integrated Processes
Observing
Comparing and
contrasting
Relating
Paper
Two squares per model in
one or two colors
Additional Materials
Bird tetrahedron models
Creator
Tomoko Fusè
PAPER SQUARE GEOMETRY
Management
1. Because this activity asks students to draw
comparisons between the bird tetrahedron and
the regular tetrahedron it is very important that
both models be made from the same size paper
so that the comparisons are valid.
Focus
Students will fold a
regular tetrahedron
and compare its
properties to those
of the bird
tetrahedron.
Procedure
1. Hand out the folding instructions (pages 33-37) and
two squares of paper to each student. Guide
students through the construction of the tetrahedron unit step by step.
2. Have students fold the second unit individually, being sure that they
made it a mirror image of the first unit. Take the class through the
assembly process, giving assistance as needed.
3. When all students have successfully assembled their tetrahedron, hand
out the remaining student sheet. Students should also have their bird
tetrahedron models on hand.
4. Have students work together in small groups to answer the questions and
compare the two models.
5. Close with a time of class discussion where students share the discoveries
they made about the properties of their tetrahedron and how these relate
to the bird tetrahedron.
Discussion
1. How many faces does a regular tetrahedron have? [four] What shape
are they? [equilateral triangles]
2. How does this compare to the number and shape of the faces on the bird
tetrahedron? [The bird tetrahedron has six faces that are isosceles right triangles.]
3. How many vertices does a regular tetrahedron have? [four] How many
edges? [six] How do these values compare to those for the bird
tetrahedron? [The bird tetrahedron has five vertices and nine edges.]
4. What was your group’s plan for finding the surface area of the tetrahedron?
5. If the base of one face is 3 units and the height is 2.5 units, what is the
surface area of the entire tetrahedron? [The surface area of one face is
3.75 units, making the surface area of the entire tetrahedron 15 units2.]
6. How does this surface area compare to the surface area that you
calculated for the bird tetrahedron? [The surface area of the bird
tetrahedron is nine units2.]
7. Were you surprised that the polyhedron with more faces had a smaller
surface area? Why or why not?
8. How can you explain this apparent paradox? [While the bird tetrahedron has
two more faces than the regular tetrahedron, the surface area of each face is
more than two units less than the surface area of each face of the tetrahedron.
This accounts for the difference in total surface area of the two polyhedra.]
Extensions
1. Have students complete the Equilateral Triangle Exploration.
2. Challenge students to construct a square-based pyramid by using two
identical tetrahedron units and the flat square unit from the Tetrahedron
Puzzle activity. This can be compared and contrasted with the tetrahedron.
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© 2000 AIMS Education Foundation
1. Fold the square in half vertically and unfold.
2. Fold from the bottom left corner as indicated by the dashed line so that the bottom right corner
touches the midline.
3. Fold the right side over so that the two points marked with dots meet as shown.
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© 2000 AIMS Education Foundation
4. Unfold completely and fold the paper horizontally so that the two points marked by dots meet. The
horizontal fold should go through the intersection of the two diagonals.
5. a. Fold the top part of the paper down at the point where the bottom edge meets the paper.
b. Unfold the bottom half, but leave the top part folded down.
a.
b.
6. Crease as indicated by each of the dashed lines, bringing the corners in to meet the horizontal
midline.
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© 2000 AIMS Education Foundation
7. Fold the top left and bottom right corners where indicated by the dashed lines so that the corners
touch the nearest diagonals. Notice that the two new sides formed are parallel to the nearest
diagonals.
8. Fold again along the diagonals so that the two sides meet in the center.
9. Flip the paper over and crease where indicated by the dashed lines.
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© 2000 AIMS Education Foundation
10. Repeat steps one through six with the second square. Steps seven through nine will be done as
mirror images.
Fold the top right and bottom left corners where indicated by the dashed lines so that the corners
touch the nearest diagonals. Notice that the two new sides formed are parallel to the nearest
diagonals.
11. Fold again along the diagonals so that the two sides meet in the center.
12. Flip the paper over and crease where indicated by the dashed lines.
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© 2000 AIMS Education Foundation
Connect the pieces as shown, folding the units so that each point is inserted into the indicated
pocket. You should be left with a regular tetrahedron.
Top view
PAPER SQUARE GEOMETRY
Side view
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© 2000 AIMS Education Foundation
Use your completed tetrahedron model to answer the following questions.
1. How many faces does a regular tetrahedron have? What shape are they?
2. How does this compare to the number and shape of the faces on a bird
tetrahedron?
3. How many vertices does a regular tetrahedron have? How many edges?
How do these values compare to those for the bird tetrahedron?
4. How would you find the surface area of one face of the tetrahedron? …of
the whole tetrahedron? Describe your plan below.
5. If the base of one face is 3 units and the height is 2.5 units, what is the
surface area of the tetrahedron?
6. How does this surface area compare to the surface area you calculated for the bird
tetrahedron?
7. Does the difference in these two values surprise you? Why or why not?
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© 2000 AIMS Education Foundation