Transcript Folding Polygons From a Circle

Folding Polygons
From a Circle
A circle cut from a regular sheet of typing paper is
a marvelous manipulative for the mathematics
classroom. Instead of placing an emphasis on
manipulating expressions and practicing
algorithms, it provides a hands-on approach fro
the visual and tactile learner.
1. Mark the center of your circular disk with a pencil. Fold
the circle in half. What is the creased line across the
disk called? Fold in half again to determine the true
center. What are these two new segments called?
What angle have you formed? Unfold the circle. How
many degrees are there in a circle? Was your
estimate of the center of the circle a good one?
Compare your result with that of your neighbor.
Vocabulary
plane
circle
arc of a circle
degrees in a circle
semicircle
degrees in a semicircle
center of circle
diameter
endpoint
line segment
midpoint of a line segment
radius
2. Place a point on the circumference of the
circle. Fold the point to the center. What is
this new segment called?
Vocabulary
circumference of a circle
area of a circle
chord
3. Fold again to the center, using one
endpoint of the chord as an endpoint for
your new chord.
Vocabulary
sector of a circle
4. Fold the remaining arc to the center.
What have you formed? Compare your
equilateral triangle with that of your
neighbor. Throughout of the rest of this
activity suppose that the area of your
triangle is one unit.
Vocabulary
area of a triangle = 1/2 base x height
triangle
equilateral triangle
isosceles triangle
equiangular triangle
sum of the measures of the angles in a triangle = 180
degrees
base
vertex
point
altitude
mediancircumcenter
incenter
Vocabulary
orthocenter
centroid
angle bisector
perpendicular bisector
perimeter of a triangle
scalene triangle
right triangle
hypotenuse
legs of a right triangle
special 30-60-90 degree triangle
Pythagorean theorem
triangle inscribed in a circle
5. Find the midpoint of one of the sides of
your triangle. Fold the opposite vertex to
the midpoint. What have you formed?
What is the area of the isosceles
trapezoid if the area of the original
triangle is one unit?
Vocabulary
trapezoid
parallel vs not parallel sides
isosceles trapezoid
area of a trapezoid = 1/2 height (top base + bottom
base)
quadrilateral
fractions
rectangle
right angle
area of a rectangle = length x width
perimeter of a rectangle
6. Notice that the trapezoid consists of three
congruent triangles. Fold one of these
triangles over the top of the middle
triangle. What have you formed? What
is its area?
Vocabulary
parallelogram
parallel lines
area of a parallelogram
polygon
regular polygon
perimeter of any polygon
rhombus
area of a rhombus
length
7. Fold the remaining triangle over the top
of the other two. What shape do you
now have? What is its area? The
triangle is similar to the unit triangle we
started with.
Vocabulary
similar
congruent
8. Place the three folded over triangles in
the palm of your hand and open it up to
form a three dimensional figure. What
new shape have you made? What is its
surface area?
Vocabulary
pyramid
surface area
faces
base
edge
9. Open it back up to the large equilateral
triangle you first made. Fold each of the
vertices to the center of the circle. What
have you formed? What is its area?
Vocabulary
hexagon
pentagon
central angles of polygons
sum of the measures of the interior angles
of a polygon
10. Turn the hexagon over and with a
crayon, pen, or pencil shade the
hexagon. Remember what the area of
this hexagon is when compared to the
original equilateral triangle. Turn the
figure over again. Push gently toward
the center so that the hexagon folds up
to form a truncated tetrahedron. What is
its surface area?
Vocabulary
tetrahedron
platonic solid
truncated tetrahedron
11. Using only the fold lines already
determined, create different polygonal
figures and determine their area. Using
only the existing fold lines, can you
construct figures with the following
areas? Are there any others? If so,
sketch them.
1 , 1 , 19 , 2 , 3 , 7 , 8 , 7 , 23
4 2 36 3 4 9 9 18 36
Vocabulary
fractions
12. You can tape twenty truncated tetrahedra
together to make an icosahedron. There will
be five on the top, five on the bottom, and
ten around the middle.
Vocabulary
icosahedron
Other vocabulary that might be used:
common denominator
arithmetic of fractions
closed set
bounded set
compact set
interior of a set
quadrants
secant line
Euler Line