What is a Paper Cup?
......

A paper cup is a cup which you can fold from a sheet of paper.
If the paper is not printed inside and sturdy, you can even drink from
it. 
If the paper cup is large enough, you can turn it and use it as a hat.
Folding of a Paper Cup top
1
......

Take a sheet of paper of the size A4 (8 1/2" x 11"). Paper used for
ink jet printers will do.
Fold the upper half down on the red line.

2
...

Cut off the lower strip.

3
......

The result is a square.

4
...... 
Fold the upper triangle downwards on the red line. 
5

The result is a isosceles, rightangled triangle.

6

Fold the left corner upwards on the red line. Mind, that you get a
new isosceles, rightangled triangle (grey) and that its basic side is
parallel to the basic side of the whole triangle.

7

It must look like this.

8

Fold the right corner upwards on the red line.

9
......

It must look like this.

10

Fold the upper triangle downwards on the red line.

11

It must look like this. There is a second triangle.

12

Fold this triangle backwards.

13
............

Open it. The paper cup is finished. It has no stand.

Variation top
...... 
After step 1 to 7 fold the left triangle backwards at the blue line.
Both triangles, in front and on the rear side, form pockets. You can
put in the upper triangles. You can cut them, so that they fit. Then the
paper cup looks better. 
Some Mathematics top
If you unfold the paper, you can recognize the
shapes of the cup in the shape of a trapezoid.
...... 
How can you calculate the measurements of the cup, when the side of
the square is given as a?
A trapezoid is given by the parallel sides x, y and the height z for
instance. 
If you like to calculate yourself, stop reading further.

If you fold on the red line, you lay the left yellow triangle on the
right. Thus the yellow quadrilateral has the sides b and b' twice. Both
diagonals are perpendicular to each other. Therefore the quadrilateral
is a rhombus and all sides have the same length. The upper triangle is
isosceles and rightangled. Its two sides have the length sqr(2)/2*b.
If the side of the starting square is a (green), there is a=b + sqrt(2)/2*b.
Then b is [2sqrt(2)]*a.


The trapezoid, which develops, is given by x, y and z.
There is:
x=sqrt(a)2b = [3*sqrt(2)4]*a
y=b=[2sqr(2)]*a
z=[sqr(2)/2]*a  b/2 = [sqr(2)1]*a

The median is m=(x+y)/2=[sqrt(2)1]*a =z. Thus the median and the height
of the trapezoid have the same lengths.
By the way the upper corners of the trapezoid devide the side a in
the ratio 1:sqrt(2). This is the ratio of the format A.
Paper Cup on the Internet
top
German:
Andreas Bauer
Becher
 traditionell
Labbé (zzzebra)
Trinkbecher
aus Papier
Mathekiste
Symmetrie
und Papierfalten
origami.ch
Trinkbecher
English:
Montessori World
Paper
Cup
OrigamiClub
Cup,
Animation
ViewDo LLC
Paper cup
(Video)
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematischebasteleien.de/
©
2003 Jürgen Köller
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