# New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics

## Authors

• Sayed El-Mongy Egypt Nuclear Regulatory Authority

## Keywords:

astronomical distance., arc length, Is constant, (eπr sA) formula, : S. El-mongy`s theorem

## 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.

## References

David Joice, (2013) “Euclids Elements Book I”,”Department of Mathematics and Computer Science, Clark University, Worcester, MA 01610.

Pat Collingwood, Ruth Emond &Rona Woodward (2007).”The Theory Circle: A Tool for Learning and for Practice”, journal Social Work Education. https://doi.org/10.1080/02615470601141409 ,P 70-83 ,

The circle, sphere and astronomy websites; www.wikipedia.org. updated on 2020

Astronomy in the Quran. www.islam-guide.com › bqs ›.

Andrew Norton, (2016). “An Introduction to Astrophysics and Cosmology”. http://www.open.ac.uk/science/physical-science/sites/www.open.ac.uk.science.physical-science/files/files/book.pdf.

R.D. Sarva Jagannadha Reddy (2014) “An Alternate Formula in terms of Pi to find the Area of a Triangle and a Test to decide the True Pi value (Atomic Energy Commission Method”, IOSR Journal of Mathematics, e-ISSN: 2278-5728, Volume 10, Issue 4.

Monica Young,(2020).”Astronomers Discover Huge Circular Arc near the Big Dipper”, https://skyandtelescope.org/astronomy-news/astronomers-discover-huge-circular-arc-near-the-big-dipper/.

Tumlinson et al., (2018). “Annual Reviews of Astronomy & Astrophysics”, .USA, SJR, Annual review Inc. ISSN 15454282, 00664146.

Kim Plof'ker, (2002). “Spherical Trigonometry and the Astronomy of the Medieval Kerala School”. Brown University,https://link.springer.com/chapter/10.1007%2F978-94-015-9862-08

Howard E. Bond, Edmund P. Nelan, Nancy Remage Evans , Gail H. Schaefer , and Dianne Harmer,(2018). “Hubble Space Telescope Trigonometric Parallax of Polaris B, Companion of the Nearest Cepheid”. The Astrophysical Journal, 853:55 (8pp),

Sky and telescope eBook,(2006).” Black Holes: Spinning hearts of darkness light the universe”.

2020-11-20

## How to Cite

El-Mongy, S. (2020). New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics . JOURNAL OF ADVANCES IN PHYSICS, 18, 158–163. https://doi.org/10.24297/jap.v18i.8914

Articles