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. ${\ displaystyle \ pi}$

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 . ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$

construction

Kochański approximation 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 . ${\ displaystyle r \ cdot \ pi}$

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: ${\ displaystyle \ pi}$

${\ displaystyle {\ overline {AZ}} = r \ cdot {\ sqrt {(3- \ tan 30 ^ {\ circ}) ^ {2} +4 \,}} \ = \ r \ cdot {\ sqrt { {\ frac {40} {3}} - 2 \ cdot {\ sqrt {3}} \}} \ \ approx \ 3 {,} 1415333 \ cdot r}$

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

Approximate construction for the squaring of Kochański's 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 . ${\ displaystyle r \ cdot \ pi}$

The area of the circle is . ${\ displaystyle r ^ {2} \ cdot \ pi}$

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]).

[Grillmayer-1] 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]).