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Measuring angles - Degrees and Protractors
Measuring angles - Degrees and Protractors
Published: 2016/08/18
Channel: tecmath
Math Antics - Angles & Degrees
Math Antics - Angles & Degrees
Published: 2013/09/10
Channel: mathantics
❤︎² Radians and Degrees (mathbff)
❤︎² Radians and Degrees (mathbff)
Published: 2014/07/02
Channel: mathbff
Measuring angles in degrees | Angles and intersecting lines | Geometry | Khan Academy
Measuring angles in degrees | Angles and intersecting lines | Geometry | Khan Academy
Published: 2011/09/21
Channel: Khan Academy
How to draw angles 30 45 60 90 degrees with Compass
How to draw angles 30 45 60 90 degrees with Compass
Published: 2013/02/16
Channel: SuccessCDs Education
Introduction to angles using degrees  (Geometry)
Introduction to angles using degrees (Geometry)
Published: 2014/08/31
Channel: Socratica
How to Draw 220 degrees Reflex Angle
How to Draw 220 degrees Reflex Angle
Published: 2016/10/28
Channel: Anil Kumar
Use your arms with 45 degree angle - DK Yoo
Use your arms with 45 degree angle - DK Yoo
Published: 2017/09/20
Channel: DK YOO
Draw an angle of 240  300 and 360 Degree
Draw an angle of 240 300 and 360 Degree
Published: 2016/12/12
Channel: STUDY MATERIAL Channel
Learn 135 degree angle without protractor or angle tool
Learn 135 degree angle without protractor or angle tool
Published: 2017/09/03
Channel: Nalemitho Easy Drawing
Constructing an Angle of 75 degrees
Constructing an Angle of 75 degrees
Published: 2013/06/09
Channel: Antonija Horvatek
How to construct angles 105 120 135 150 degrees - Compass
How to construct angles 105 120 135 150 degrees - Compass
Published: 2013/02/16
Channel: SuccessCDs Education
Constructing an Angle of 90 degrees
Constructing an Angle of 90 degrees
Published: 2013/06/03
Channel: Antonija Horvatek
Finding 45 Degree From A Right Angle Using A Compass  - How To Draw / Bisect / Bisection
Finding 45 Degree From A Right Angle Using A Compass - How To Draw / Bisect / Bisection
Published: 2013/04/24
Channel: Savvas Papasavva
75 degree angle.avi
75 degree angle.avi
Published: 2012/05/03
Channel: O.R. Anderson
Learn to draw 75 degree angle without protractor or angle tool
Learn to draw 75 degree angle without protractor or angle tool
Published: 2017/03/06
Channel: Nalemitho Easy Drawing
How do we Construct 30, 60 and 120 Degree Angles in one go?
How do we Construct 30, 60 and 120 Degree Angles in one go?
Published: 2014/12/20
Channel: Don't Memorise
How do we Construct a 30 Degree Angle?
How do we Construct a 30 Degree Angle?
Published: 2014/12/20
Channel: Don't Memorise
How to Construct 10 degree angle easily
How to Construct 10 degree angle easily
Published: 2016/10/26
Channel: Technical Shubham
Drawing 105 degree angle without protractor or angle tool
Drawing 105 degree angle without protractor or angle tool
Published: 2017/03/07
Channel: Nalemitho Easy Drawing
Draw 15 degree angle without protractor or angle tool with compass
Draw 15 degree angle without protractor or angle tool with compass
Published: 2017/09/30
Channel: Nalemitho Easy Drawing
How to Draw 40° degree Angle with compass And scale
How to Draw 40° degree Angle with compass And scale
Published: 2017/07/01
Channel: Qurious to some Furious
Constructing an Angle of 120 degrees
Constructing an Angle of 120 degrees
Published: 2013/06/03
Channel: Antonija Horvatek
Constructing an Angle of 30 degrees
Constructing an Angle of 30 degrees
Published: 2013/06/03
Channel: Antonija Horvatek
How to construct 55 degree angle
How to construct 55 degree angle
Published: 2017/04/23
Channel: Geomatrix Guys
Constructing an Angle of 45 degrees
Constructing an Angle of 45 degrees
Published: 2013/06/03
Channel: Antonija Horvatek
Draw 150 degree angle without protractor or angle tool with compass
Draw 150 degree angle without protractor or angle tool with compass
Published: 2017/09/30
Channel: Nalemitho Easy Drawing
How to Construct 20 and 50 degree of angle
How to Construct 20 and 50 degree of angle
Published: 2017/04/02
Channel: galaxy coaching classes
Jak skonstruować kąt 65 stopni Construction of 65 degree angle
Jak skonstruować kąt 65 stopni Construction of 65 degree angle
Published: 2017/01/28
Channel: Mat hub
Watch This Hybrid Range Rover SUV Climb 999 Steps Up A 45-Degree Angle Mountain In China
Watch This Hybrid Range Rover SUV Climb 999 Steps Up A 45-Degree Angle Mountain In China
Published: 2018/02/19
Channel: Business Insider
To construct 20 degree angle
To construct 20 degree angle
Published: 2016/02/12
Channel: Technical Shubham
Learn to draw 45 degree angle without protractor or angle tool
Learn to draw 45 degree angle without protractor or angle tool
Published: 2016/10/24
Channel: Nalemitho Easy Drawing
Constructing a 22,5 degree angle
Constructing a 22,5 degree angle
Published: 2017/02/24
Channel: Geoffrey Comins
How To make 50 Degree Angle - With Compass
How To make 50 Degree Angle - With Compass
Published: 2017/03/30
Channel: Asjad Siddiqui
ANGLE CONSTRUCTIONS USING COMPASS - 135 DEGREES
ANGLE CONSTRUCTIONS USING COMPASS - 135 DEGREES
Published: 2013/04/08
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How to make 20,80,55 degree angle with compass
How to make 20,80,55 degree angle with compass
Published: 2017/12/28
Channel: AVR ENTERTAINMENT AND GUIDE
Quickest way to draw 20 degree angle by compass
Quickest way to draw 20 degree angle by compass
Published: 2016/11/13
Channel: RAVI GANGWAR
How To Draw A 30 Degree Angle With A Compass And Ruler (No Protractor)
How To Draw A 30 Degree Angle With A Compass And Ruler (No Protractor)
Published: 2013/09/26
Channel: maths3000
Constructing angle of 165 degree
Constructing angle of 165 degree
Published: 2016/11/22
Channel: Girish Rao
Learn 165 degree angle without protractor or angle tool
Learn 165 degree angle without protractor or angle tool
Published: 2017/09/03
Channel: Nalemitho Easy Drawing
New Boeing Aircraft | 90 Degree Angle Take Off
New Boeing Aircraft | 90 Degree Angle Take Off
Published: 2017/11/12
Channel: Ankit Prakash
Hard Drilling Big Hole in Stainless Steel on 45 Degree Angle Tubing
Hard Drilling Big Hole in Stainless Steel on 45 Degree Angle Tubing
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60 Degree angle cuts - by Matt C.
60 Degree angle cuts - by Matt C.
Published: 2015/02/09
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How to make 165 degree angle
How to make 165 degree angle
Published: 2017/03/05
Channel: Fast Maths
How to construct a 60 degree angle / How to construct an equilateral triangle
How to construct a 60 degree angle / How to construct an equilateral triangle
Published: 2012/01/05
Channel: maths520
How to Frame a 45 Degree Angle Wall
How to Frame a 45 Degree Angle Wall
Published: 2016/04/20
Channel: Basement Finishing Man
Constructing an Angle of 60 degrees
Constructing an Angle of 60 degrees
Published: 2013/06/03
Channel: Antonija Horvatek
How to use an angle degree torque gauge - Auto Information Series
How to use an angle degree torque gauge - Auto Information Series
Published: 2013/11/16
Channel: Robert DIY
How to construct 270 degree angle with compass
How to construct 270 degree angle with compass
Published: 2017/09/02
Channel: Techie Prince
How to make 105 degree angle with compass
How to make 105 degree angle with compass
Published: 2017/03/06
Channel: AWESOMENESSwithRohan V
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Degree
Unit system Non-SI accepted unit
Unit of Angle
Symbol °[1][2] or deg[3]
Unit conversions
1 °[1][2] in ... ... is equal to ...
   turns    1/360 turn
   radians    π/180 rad
   gons    10/9g
One degree (shown in red) and
eighty nine degrees (shown in blue)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.[4] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.

History[edit]

A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.

The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year.[5] Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.

Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.[6][7] The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree.

Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.[8][9] Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.[10] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.

The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda:[11]

Twelve spokes, one wheel, navels three.

Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.

— Dirghatamas, Rigveda 1.164.48

Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,[note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in the OEIS).[12][13] Furthermore, it is divisible by every number from 1 to 10 except 7.[note 2] This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention.

Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above.

Subdivisions[edit]

For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates (latitude and longitude), degree measurements may be written using decimal degrees, with the degree symbol behind the decimals; for example, 40.1875°.

Alternatively, the traditional sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). Use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are respectively represented by a single and double prime. For example, 40.1875° = 40° 11′ 15″, or, using quotation mark characters, 40° 11' 15". Additional precision can be provided using decimals for the arcseconds component.

The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today. These subdivisions were denoted[citation needed] by writing the Roman numeral for the number of sixtieths in superscript: 1I for a "prime" (minute of arc), 1II for a second, 1III for a third, 1IV for a fourth, etc. Hence the modern symbols for the minute and second of arc, and the word "second" also refer to this system.[citation needed]

Alternative units[edit]

A chart to convert between degrees and radians

In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = ​π180.

The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. One turn is equal to 360°.

With the invention of the metric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees"[note 3] called grad or gon, where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = ​109 gon). Although that idea was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades (​14,000) were used with French artillery sights in World War I.

An angular mil, which is most used in military applications, has at least three specific variants, ranging from ​16,400 to ​16,000. It is approximately equal to one milliradian (c.16,283). A mil measuring ​16,000 of a revolution originated in the imperial Russian army, where an equilateral chord was divided into tenths to give a circle of 600 units. This may be seen on a lining plane (an early device for aiming indirect fire artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Conversion of common angles
Turns Radians Degrees Gradians (Gons)
0 0 0g
1/24 π/12 15° 16 2/3g
1/12 π/6 30° 33 1/3g
1/10 π/5 36° 40g
1/8 π/4 45° 50g
1/2π 1 c. 57.3° c. 63.7g
1/6 π/3 60° 66 2/3g
1/5 2π/5 72° 80g
1/4 π/2 90° 100g
1/3 2π/3 120° 133 1/3g
2/5 4π/5 144° 160g
1/2 π 180° 200g
3/4 3π/2 270° 300g
1 2π 360° 400g

See also[edit]

Notes[edit]

  1. ^ The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
  2. ^ Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.
  3. ^ These new and decimal "degrees" must not be confused with decimal degrees.

References[edit]

  1. ^ HP 48G Series – User's Guide (UG) (8 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90126, (00048-90104). Retrieved 2015-09-06. 
  2. ^ HP 50g graphing calculator user's guide (UG) (1 ed.). Hewlett-Packard. 2006-04-01. HP F2229AA-90006. Retrieved 2015-10-10. 
  3. ^ HP Prime Graphing Calculator User Guide (UG) (PDF) (1 ed.). Hewlett-Packard Development Company, L.P. October 2014. HP 788996-001. Retrieved 2015-10-13. 
  4. ^ Bureau International des Poid et Mesures (2006). "The International System of Units (SI)" (8 ed.). Archived from the original on 2009-10-01. 
  5. ^ "Degree". MathWorld. 
  6. ^ Jeans, James Hopwood (1947). The Growth of Physical Science. p. 7. 
  7. ^ Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2. 
  8. ^ Rawlins, Dennis. "On Aristarchus". DIO - The International Journal of Scientific History. 
  9. ^ Toomer, Gerald J. Hipparchus and Babylonian astronomy. 
  10. ^ "2 (Footnote 24)". Aristarchos Unbound: Ancient Vision / The Hellenistic Heliocentrists' Colossal Universe-Scale / Historians' Colossal Inversion of Great & Phony Ancients / History-of-Astronomy and the Moon in Retrograde! (PDF). DIO - The International Journal of Scientific History. 14. March 2008. p. 19. ISSN 1041-5440. Retrieved 2015-10-16. 
  11. ^ Dirghatamas. Rigveda. pp. 1.164.48. 
  12. ^ Brefeld, Werner. "Divisibility highly composite numbers". 
  13. ^ Brefeld, Werner (2015). (not defined). Rowohlt Verlag. pp. Not yet published. 

External links[edit]

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