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22-572: Southwest Portland is one of the sextants of Portland, Oregon . Downtown Portland lies in the Southwest section between the I-405 freeway loop and the Willamette River, centered on Pioneer Courthouse Square ("Portland's living room"). Downtown and many other parts of inner Portland have compact square blocks (200 ft [60 m] on a side) and narrow streets (64 ft [20 m] wide),

44-487: A chord formed with the extremal points of the arc is given by C = 2 R sin ⁡ θ 2 {\displaystyle C=2R\sin {\frac {\theta }{2}}} where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector in radians. Degree (angle) A degree (in full, a degree of arc , arc degree , or arcdegree ), usually denoted by ° (the degree symbol ),

66-410: A " prime " (minute of arc), 1 for a second , 1 for a third , 1 for a fourth , etc. Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system. SI prefixes can also be applied as in, e.g., millidegree , microdegree , etc. In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This

88-540: A circle is πr . The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2 π (because the area of the sector is directly proportional to its angle, and 2 π is the angle for the whole circle, in radians): A = π r 2 θ 2 π = r 2 θ 2 {\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}} The area of

110-452: A cycle or revolution) 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" in France and nearby countries, where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = 10 ⁄ 9 gon). This was called grade (nouveau) or grad . Due to confusion with

132-429: A pedestrian-friendly combination. Many of Portland's recreational, cultural, educational, governmental, business, and retail resources are concentrated downtown, including: Beyond downtown, the Southwest section also includes: Sextant (circle) The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle. A sector with

154-1145: A sector in terms of L can be obtained by multiplying the total area πr by the ratio of L to the total perimeter 2 πr . A = π r 2 L 2 π r = r L 2 {\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}} Another approach is to consider this area as the result of the following integral: A = ∫ 0 θ ∫ 0 r d S = ∫ 0 θ ∫ 0 r r ~ d r ~ d θ ~ = ∫ 0 θ 1 2 r 2 d θ ~ = r 2 θ 2 {\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}\,d{\tilde {r}}\,d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}\,d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}} Converting

176-533: 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. The earliest trigonometry , used by the Babylonian astronomers and their Greek successors,

198-453: Is 1 nautical mile . The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25). 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 by writing the Roman numeral for the number of sixtieths in superscript: 1 for

220-513: Is a measurement of a plane angle in which one full rotation is 360 degrees. It is not an SI unit —the SI unit of angular measure is the radian —but it is mentioned in the SI brochure as an accepted unit . Because a full rotation equals 2 π radians, one degree is equivalent to ⁠ π / 180 ⁠ radians. The original motivation for choosing the degree as a unit of rotations and angles

242-511: 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. 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 ( DD notation ); for example, 40.1875°. Alternatively,

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264-442: 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 2 π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant : 1° = π ⁄ 180 . One turn (corresponding to

286-535: Is in radians. The formula for the length of an arc is: L = r θ {\displaystyle L=r\theta } where L represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle. If the value of angle is given in degrees, then we can also use the following formula by: L = 2 π r θ 360 {\displaystyle L=2\pi r{\frac {\theta }{360}}} The length of

308-458: Is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. 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 and the Babylonian calendar , used 360 days for

330-457: The OEIS ). Furthermore, it is divisible by every number from 1 to 10 except 7. 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

352-522: The central angle into degrees gives A = π r 2 θ ∘ 360 ∘ {\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}} The length of the perimeter of a sector is the sum of the arc length and the two radii: P = L + 2 r = θ r + 2 r = r ( θ + 2 ) {\displaystyle P=L+2r=\theta r+2r=r(\theta +2)} where θ

374-430: The central angle of 180° is called a half-disk and is bounded by a diameter and a semicircle . Sectors with other central angles are sometimes given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one quarter, sixth or eighth part of a full circle, respectively. The arc of a quadrant (a circular arc ) can also be termed a quadrant. The total area of

396-573: The existing term grad(e) in some northern European countries (meaning a standard degree, ⁠ 1 / 360 ⁠ of a turn), the new unit was called Neugrad in German (whereas the "old" degree was referred to as Altgrad ), likewise nygrad in Danish , Swedish and Norwegian (also gradian ), and nýgráða in Icelandic . To end the confusion, the name gon was later adopted for

418-403: The first Greeks known to divide the circle in 360 degrees of 60 arc minutes . Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Another motivation for choosing the number 360 may have been that it is readily divisible : 360 has 24 divisors , making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in

440-481: The new unit. Although this idea of metrification was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades ( 1 ⁄ 4,000 ) 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 1 ⁄ 6,400 to 1 ⁄ 6,000 . It

462-588: 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 represented by a single prime (′) and double prime (″) respectively. For example, 40.1875° = 40° 11′ 15″ . Additional precision can be provided using decimal fractions of an arcsecond. Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude

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484-432: 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. Timocharis , Aristarchus, Aristillus , Archimedes , and Hipparchus were

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