Abstract
The circle and sphere have been studied since the ancient Egyptians and Greeks before the Common Era (BCE). The recent scientific renaissance has also used them in different fields. It is also mentioned in the Prophet Mohamed`s Holy Quran. This article introduces a new Theorem (S. El-Mongy`s Theorem) as an empirical formula to correlate the constant (e) with circle and sphere. It states that “the arc length is correlated as a direct function in {(e π r sA)}, whatever the central angle (ϴ) and radius (r). The factor sA is (ϴ/10ϕ). The formula can also be written as; AL = {(0.0174533185 r ϴ)}. Where the value 0.0174533185 is a constant called Sayed`s number (Is) and equals (eπ/10ϕ). This factor is very close to value (π/180 = 0.0174532925) with ~1.5x10-4 % difference. The formula was applied for calculation the arc length (AL) of circles of different radii and angles. The results of this formula were validated and verified for very wide range; from 0.5 cm to 4.4x1023 km (46.5x109 light-years; ly) and compared with the old published arc length formula results. The difference is from 0.000% to 0.002% only. The formula was also used as trigonometric functions of circular orbits for calculation the distances between the Earth and Sun, Moon, planets, stars, and EH-M87 Black hole with relatively small error; the difference is from 0.26% to maximum ~ 2.27%. The error was 0.29% for ~54 x 106 ly distance to the M87 black hole. The S. El-Mongy formula may open the door for further scientific and engineering applications.
Highlights
The circle has been known since before the beginning of recorded history
A circle is a plane figure bounded by one curved line and all the bounding line is equal as given by Euclid, Elements Book, 300 before the Common Era (BCE) (1,3)
The arc length can be calculated using our abovementioned empirical formula. It was validated for different circle radii (0.5 cm – 4.4x1023 km) and central angles Θ (0.00050 – 3600)
Summary
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon and Sun. El-mongys Theorem: “The circle arc length is a function in (e π r sA)”. A new empirical formula for calculation of the circle arc length (AL) was introduced in this article. The old formula for arc length can be given as follows (2,3): The circle circumstance = 2 π r
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