Circles space all similar, and also "the circumference divided by the diameter" to produce the same value regardless of your radius. This worth is the ratio of the circumference of a circle to its diameter and is referred to as π (Pi). This consistent appears in the calculation of the area that a circle, and is a type of an irrational number recognized as a transcendental number that deserve to be expressed neither by a portion nor by any type of radical authorize such together a square root, nor your combination. The number has actually an infinite variety of decimal places, namely, 3.1415926535..., and it has actually now been computed to 5 trillion decimal locations by computers.
You are watching: Relationship between circumference and diameter of a circle
As because that the value of π, ancient civilizations provided their own. Together a regular hexachathamtownfc.netn that is inscribed in a circle v a radius of 1 has actually a perimeter of 6, the is revealed that Pi has actually a value better than 3. In the old Egypt, they acquired an approximation of
by place a constant octachathamtownfc.netn ~ above a circle, and in old Babylonia they used
Archimedes pertained to the conclusion in his work-related Kyklu metresis (measure the a circle) that Pi satisfies
In Europe, Viete (1540-1603) discovered the an initial formula the expresses π:
After that, the Wallis (1616-1703) Formula:
the Grechathamtownfc.netry (1638-1675) and also Leibniz (1646-1716) Formula:
Moreover, Newton (1642-1727) and Euler (1707-1783) discovered a series that converged faster, which enabled them to calculate worths of Pi to an ext decimal places. If we usage the relation
we can achieve a worth of 3.14159 because that π precise to five decimal places with the an initial 4 regards to the Taylor development of tan-1.In a recent computer calculation, the adhering to equations to be used:
* tan-1 : Arc tangent. The inverse duty of tangent.
Calculation that Pi in Wasan
At the end of Sanpo shojo, a an approach for calculating Pi appears. To summary what is explained in the book, the an approach is as follows: i think the original number=3,
Continue this until difference 100 is created. Then, Pi is obtained by adding the original number, difference 1, distinction 2, distinction 3...and for this reason on. Rewriting this together a math expression, that is presented to have following regularity:
When incrementing n for
holds true; Hasegawa uses this to attain the an outcome of
We perform not recognize anything around the number"s regularity from this an outcome alone. In fact, however, over there is a relationship between the terms. Every term is chose by multiplying its previous term by a regular portion as follows:
Kikuchi noticed that such a collection was what K. F. Gauss (1777-1855) called a hypergeometric series. A hypergeometric series is characterized as follows:
Therefore, Kikuchi verified in the next record that the calculate
in Enri shinko by Wada Yasushi was tantamount to
was indistinguishable to
and that Matsunaga"s
was identical to
In Wasan, Seki Takakazu, Takebe Katahiro, etc., seek calculation formulas for π2.
, acquired by Takebe, is the very first formula to evaluate Pi in the history of Wasan. Takebe calculated π come 41 decimal areas with this formula. In the following treatise, Kikuchi derived
to to express the square of s or the arc of a circle with sagitta c and also diameter d, i m sorry was explained by Yamaji Nushizumi in Kenkon no maki (c 1765), and proved
as soon as
Arc, Sagitta, and also Diameter of a Circle
See more: How Did The Geography Of Greece Affect Its Development ? How Did Geography Affect Greece Development
In the critical paper, he showed that
derived by Ajima Naonobu in Kohai jutsukai might be streamlined to
In the following year, Kikuchi also wrote a record to present a technique of calculating the length of an arc derived by Takebe Katahiro in Tokyo Sugaku Butsuri Gakkai Kiji Vol. 8 (1897). This collection of papers was intended to introduce to the people the fact that theories he had found in calculations of Pi in Wasan were equivalent to calculus in the West.