Approximate construction by Kochański
The approximate construction of Kochański is a method for determining the circle number . It is named after the Polish mathematician Adam Adamandy Kochański , who developed the construction in 1685.
Constructing a line with the length of the number is very simple: Draw a circle with radius 1 (unit circle); half the circumference is then the length . However , it is impossible to construct the number on a straight line because one of the transcendent numbers is that can not be constructed . Kochansi's approximation construction provides a very good approximation for or any multiple thereof and can also be used as part of an approximation construction for the square of the circle .
construction
- Draw a unit circle with the radius r = 1 around the center M.
- Then you draw two circular diameters that are perpendicular to one another and intersect the circular line at points A, B and C.
- From point B, you mark off the radius r on the circular line and you get point Y.
- The straight line MY intersects the circular tangent running through C at point X.
- From point X, you mark off the radius r three times on the tangent and you get point Z.
The length AZ of the (red) segment [AZ] is a very good approximation for half the circumference or for the product .
Estimation of the error
The value for determined with this approximation construction is a bit too small, differs from the actual value 3.1415926 ... but only in the fifth place after the decimal point. As can be easily calculated, the following applies:
The value determined with the approximation construction is approx. 99.99811 percent of the actual value. The error is therefore less than 2/1000 percent, or to put it another way: Only from a circle radius of r = 16.86 meters does the error of the distance AZ amount to more than one millimeter.
Use when squaring the circle
The squaring of the circle - i.e. the construction of a square of the same area from a given circle with a ruler and compass - is impossible. According to Kochański, however, the route length AZ gives us a very good approximation for the product .
The area of the circle is .
So a rectangle (here drawn in red) over the line [AZ] with the height r has almost the same area as the given circle. This rectangle can in turn be transformed into a square with the same area without errors using the method of squaring the rectangle . The square constructed in this way is a very good approximation for the unsolvable problem.
Estimation of error: The area of the yellow square is approximately 99.99811 percent of the area of the given circle. Or to put it another way: For circles with a radius smaller than 12.99 cm, the difference between the two areas is less than a square millimeter.
Individual evidence
- ↑ ^{a } ^{b} Dieter Grillmayer: In the realm of geometry: Part I: level geometry . 2. Kochanski's approximation construction. 2009, p. 49 ( books.google.de [accessed February 19, 2020]).