The paper we usually use has a standar size. In lots of countries in the world (but not in North America) we use paper size standars based in ISO 216 and we use world like DIN A0, DIN A1, DIN A2, DIN A3, DIN A4 an so on.
The base DIN A0 size of paper is defined to have an area of one square meter, and successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension. The objective is that this parts again have the same aspect ratio.
We can calculate this aspect ratio:
The aspect ratio verifies (these two rectangles are similar):
Then the larger side is equal to the diagonal of a square of size the shorter side:
DIN A0 size has one square meter. We can calculate his dimensions (rounded to milimeters)
In a photocopier, when we want to reduce from A3 to A4 the display shows a ratio of 71%. ¿Why?
I have used this proportion in the animation about the sum of the geometric series of ratio 1/2.
The doors of this piece of furniture are in the same proportion. It has been designed and made by Roberto Cardil using pine and oak wood. You can see another furniture with the golden spiral.
This proportion is different than the golden proportion.
Now we are going to see more facts about this rectangle.
Remember that the diagonal of a square is:
Then we can find our rectangle as a section of a cube:
One diagonal of this rectangle:
You can calculate D as a basic application of the Pythagorean Theorem:
If we consider the two diagonals of this section, the point of intersection is the center of the cube:
Now we are going to study the angles between the two diagonals (we need some basic knowledge about trigonometry):
Angle C is easy to calculate:
We are going to meet these two angles when we study the chamfered cube and the rhombic dodecahedron because our rectangle is related with these polyhedra.
Another approach to find angle A:
Can you explain it?
First, lines PR and QT are perpendicular because if we rotate 90 degrees counterclockwise the rectangle ....
Second, lines PR and QT are two medians of triangle QRS, then the centroid divide ...
Now you can write cosA ...