Folding Circles
Materials:
 A cutout paper circle, 7 to 10 inches in diameter. A template
is provided on the last page.
1. Fold the edge of the circle so that one of the points on the
circumference of the circle lands on the center of the circle.
This fold line defines a chord of the circle.
2. Fold a second point to the center. The resulting shape will
look like and ice-cream cone.
3.
Fold the third point to the center.
 What is the name of the shape that has been formed?
 What properties does this shape have?
 Prove that this is an equilateral triangle without using a
ruler.
4. Find the midpoint of one side of the triangle by folding one
vertex of the triangle on top of a second vertex and pinching.
Then fold the third vertex to this midpoint.
 What is the name of the shape that is formed?
 What properties does this shape have?
 If you open the triangle back up, what do you notice
about this new fold line, its endpoints, and its relationship
to the opposite side of the triangle?
 If you fold the trapezoid back, how are the tow bases of
this trapezoid related?
 If we let the area of the original triangle be 1, what is the
area of the trapezoid? How do you know?
5. Fold the vertex on the bottom right to the vertex on the top
left.
 What is the name of this new shape?
 What properties does this shape have?
 If we let the area of the original triangle be 1, what is the
area of the rhombus? How do you know?
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March 2010
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6. Now fold the vertex on the bottom left to the vertex on the
top right.
 What shape has now been formed?
 How does this new shape compare to the original triangle
from step 3? (Note: Have half the class unfold the shape
back to the original triangle so that students can compare
the two triangles.)
 What is the ratio of the length of the side of the large
triangle to the length of the side of the small triangle?
 What is the ratio of the area of the large triangle to the
area of the small triangle?
 If the area of the original triangle is 1, what is the area of
this small triangle? How do you know?
7. Unfold the shape back to the original triangle. Now fold the
three vertices of the original triangle up to a single vertex.
 What shape has been formed?
 What properties does this shape have?
 If the area of the original triangle is 1, what is the surface
area of this pyramid?
8. Unfold back to the original triangle. Choose a vertex of this
triangle and fold it to the center of the circle. Fold the other
two vertices of the triangle to the center of the circle.
 What shape do you have now? Is it regular?
 If the area of the original triangle is 1, what is the area of
the hexagon?
 If you look carefully you will see a triangle inscribed within
the hexagon. This triangle is surrounded by three small
trapezoids. What is the area of one of these trapezoids?
9. Fold the top halves of the triangles down across each other
to make a truncated pyramid.
 If the area of the original triangle is 1, what is the surface
area of this truncated pyramid? How do you know?
 Look at the two triangular bases. What is the scale
factor?
 What is the ratio of the area of the small triangle to the
area of the large triangle?
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March 2010
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10. Using only the fold lines you have already created, make
polygons that have each of the following areas. Sketch each
polygon. Let the area of the original triangle be 1.
1 3 1 2 3 8 7 5
 4, 4, 2, 3, 8, 9 ,
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March 2010
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,
36
19 23
,
36
,
36
7
18
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March 2010
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