Circular sector

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 quarter, sixth or eighth part of a full circle, respectively. The arc of a quadrant (a circular arc) can also be termed a quadrant.

{{see also|Circular arc#Sector area}}

The total area of a circle is {{math|πr{{isup|2}}}}. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle {{mvar|θ}} (expressed in radians) and {{math|2π}} (because the area of the sector is directly proportional to its angle, and {{math|2π}} is the angle for the whole circle, in radians):

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

The area of a sector in terms of {{mvar|L}} can be obtained by multiplying the total area {{math|πr{{isup|2}}}} by the ratio of {{mvar|L}} to the total perimeter {{math|2πr}}.

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

Another approach is to consider this area as the result of the following integral:

$$A\; =\; \backslash int\_0^\backslash theta\backslash int\_0^r\; dS\; =\; \backslash int\_0^\backslash theta\backslash int\_0^r\; \backslash tilde\{r\}\backslash ,\; d\backslash tilde\{r\}\backslash ,\; d\backslash tilde\{\backslash theta\}\; =\; \backslash int\_0^\backslash theta\; \backslash frac\; 1\; 2\; r^2\backslash ,\; d\backslash tilde\{\backslash theta\}\; =\; \backslash frac\{r^2\; \backslash theta\}\{2\}$$

Converting the central angle into degrees gives{{Cite book | last = Uppal | first = Shveta | title=Mathematics: Textbook for class X | date = 2019 | publisher = National Council of Educational Research and Training | isbn=978-81-7450-634-4 | location = New Delhi | pages = [http://www.ncert.nic.in/ncerts/l/jemh112.pdf#page=4 226], [http://www.ncert.nic.in/ncerts/l/jemh112.pdf#page=5 227] | oclc=1145113954}}

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

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

$$P\; =\; L\; +\; 2r\; =\; \backslash theta\; r\; +\; 2r\; =\; r\; (\backslash theta\; +\; 2)$$

where {{mvar|θ}} is in radians.

The formula for the length of an arc is:{{Cite book | last1 = Larson | first1 = Ron | author1-link = Ron Larson | first2 = Bruce H. | last2 = Edwards | url = https://www.worldcat.org/oclc/706621772 | title = Calculus I with Precalculus | date = 2002 | isbn = 978-0-8400-6833-0 | edition = 3rd | location=Boston, MA. | oclc=706621772 | publisher = Brooks/Cole | page = 570}}

$$L\; =\; r\; \backslash theta$$

where {{mvar|L}} represents the arc length, r represents the radius of the circle and {{mvar|θ}} represents the angle in radians made by the arc at the centre of the circle.{{Cite book | last = Wicks | first = Alan | url = https://www.worldcat.org/oclc/58869667 | title = Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus | date = 2004 | publisher = Infinity Publishing.com | isbn = 0-7414-2141-0 | location = West Conshohocken, PA | oclc = 58869667 | page = 79}}

If the value of angle is given in degrees, then we can also use the following formula by:{{sfnp|Uppal|2019}}

$$L\; =\; 2\; \backslash pi\; r\; \backslash frac\{\backslash theta\}\{360\}$$

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

$$C\; =\; 2R\backslash sin\backslash frac\{\backslash theta\}\{2\}$$

where {{mvar|C}} represents the chord length, {{mvar|R}} represents the radius of the circle, and {{mvar|θ}} represents the angular width of the sector in radians.

• Circular segment – the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
• Conic section
• Earth quadrant
• Hyperbolic sector
• Sector of (mathematics)
• Spherical sector – the analogous 3D figure

{{Reflist}}

• {{cite book |last= Gerard |first= L. J. V. |date= 1874 |title= The Elements of Geometry, in Eight Books; or, First Step in Applied Logic |location= London |publisher= Longmans, Green, Reader and Dyer |url= https://archive.org/details/elementsgeometr00geragoog/page/n309 |page= 285}}
• {{cite book |last= Legendre |first= Adrien-Marie |date= 1858 |title= Elements of Geometry and Trigonometry |author-link= Adrien-Marie Legendre |editor-last= Davies |editor-first= Charles |editor-link= Charles Davies (professor) |location= New York |publisher= A. S. Barnes & Co. |url= https://books.google.com/books?id=pFYliSRwxEgC&pg=RA1-PA119 |page= 119}}

Category:Circles