Jump to content

Main menu Navigation ●Main page ●Contents ●Current events ●Random article ●About Wikipedia ●Contact us ●Donate Contribute ●Help ●Learn to edit ●Community portal ●Recent changes ●Upload file

●Create account ●Log in ●Create account ● Log in Pages for logged out editors learn more ●Contributions ●Talk

(Top) 1 Types 1.1 Compass 2 Area 3 Perimeter 4 Arc length 5 Chord length 6 See also 7 References 8 Sources

Circular sector

●Afrikaans ●العربية ●Azərbaycanca ●Беларуская ●Català ●Чӑвашла ●Čeština ●ChiShona ●Dansk ●Deutsch ●Eesti ●Español ●Esperanto ●Euskara ●فارسی ●Français ●Galego ●한국어 ●Հայերեն ●Italiano ●עברית ●Latviešu ●Lëtzebuergesch ●Lietuvių ●Magyar ●Македонски ●മലയാളം ●Nederlands ●日本語 ●Norsk bokmål ●Norsk nynorsk ●Oʻzbekcha / ўзбекча ●ភាសាខ្មែរ ●Piemontèis ●Polski ●Português ●Română ●Русский ●Shqip ●Slovenščina ●کوردی ●Suomi ●Svenska ●தமிழ் ●ไทย ●Українська ●Tiếng Việt ●吴语 ●粵語 ●中文 Edit links ●Article ●Talk ●Read ●Edit ●View history Tools Actions ●Read ●Edit ●View history General ●What links here ●Related changes ●Upload file ●Special pages ●Permanent link ●Page information ●Cite this page ●Get shortened URL ●Download QR code ●Wikidata item Print/export ●Download as PDF ●Printable version In other projects ●Wikimedia Commons Appearance From Wikipedia, the free encyclopedia (Redirected from Quadrant (circle))

The minor sector is shaded in green while the major sector is shaded white.

Acircular sector, also known as circle sectorordisk sector or simply a sector (symbol: ⌔), is the portion of a disk (aclosed region bounded by a circle) enclosed by two radii and an arc, with the smaller area being known as the minor sector and the larger being the major sector.^[1] In the diagram, $θ$ is the central angle, $r$ the radius of the circle, and $L$ is the arc length of the minor sector.

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.^[2]

Types[edit]

A sector with 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 4th, 6th or 8th part of a full circle, respectively. The arc of a quadrant (acircular arc) can also be termed a quadrant.

Compass[edit]

An 8-point windrose

Traditionally wind directions on the compass rose are given as one of the 8 octants (N, NE, E, SE, S, SW, W, NW) because that is more precise than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to allow more precise indication.

The name of the instrument "octant" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the compass rose.

Area[edit]

The total area of a circle is $πr 2$ . 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=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}

A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}

The area of a sector in terms of L can be obtained by multiplying the total area $π$ r² by the ratio of L to the total perimeter 2 $π$ r.

A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}

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=\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}}

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 the central angle into degrees gives^[3]

A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}

A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}

Perimeter[edit]

The length of the perimeter of a sector is the sum of the arc length and the two radii:

P=L+2r=\theta r+2r=r(\theta +2)

P=L+2r=\theta r+2r=r(\theta +2)

where

θ

is in radians.

Arc length[edit]

The formula for the length of an arc is:^[4]

L=r\theta

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.^[5]

If the value of angle is given in degrees, then we can also use the following formula by:^[3]

L=2\pi r{\frac {\theta }{360}}

L=2\pi r{\frac {\theta }{360}}

Chord length[edit]

The length of a chord formed with the extremal points of the arc is given by

C=2R\sin {\frac {\theta }{2}}

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.

References[edit]

^ Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-8173358371.

^ Achatz, Thomas; Anderson, John G. (2005). Technical shop mathematics. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376. ISBN 978-0831130862. OCLC 56559272.

^ ^a ^b Uppal, Shveta (2019). Mathematics: Textbook for class X. New Delhi: National Council of Educational Research and Training. pp. 226, 227. ISBN 978-81-7450-634-4. OCLC 1145113954.

^ Larson, Ron; Edwards, Bruce H. (2002). Calculus I with Precalculus (3rd ed.). Boston, MA.: Brooks/Cole. p. 570. ISBN 978-0-8400-6833-0. OCLC 706621772.

^ Wicks, Alan (2004). Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus. West Conshohocken, PA: Infinity Publishing.com. p. 79. ISBN 0-7414-2141-0. OCLC 58869667.

Sources[edit]

Gerard, L. J. V., The Elements of Geometry, in Eight Books; or, First Step in Applied Logic (London, Longmans, Green, Reader and Dyer, 1874), p. 285.
Legendre, A. M., Elements of Geometry and Trigonometry, Charles Davies, ed. (New York: A. S. Barnes & Co., 1858), p. 119.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Circular_sector&oldid=1228842917#Quadrant" Category: ●Circles Hidden categories: ●Articles with short description ●Short description matches Wikidata ●This page was last edited on 13 June 2024, at 14:01 (UTC). ●Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. ●Privacy policy ●About Wikipedia ●Disclaimers ●Contact Wikipedia ●Code of Conduct ●Developers ●Statistics ●Cookie statement ●Mobile view