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Circular sector - wikipedia Circular sector A circular sector is shaded in green A circular sector or circle sector ( symbol : ⌔ ) , is the portion of a disk enclosed by two radii and an arc , where the smaller area is known as the minor sector and the larger being the major sector . In the diagram , θ is the central angle in radians , r ( \ displaystyle r ) the radius of the circle , and L ( \ displaystyle L ) is the arc length of the minor 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 , these include quadrants ( 90 ° ) , sextants ( 60 ° ) and octants ( 45 ° ) , which come from the sector being one 4th or 6th or 8th part of a full circle , respectively . 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 . Contents 1 Area 2 Perimeter 3 Arc Length 4 Chord length 5 See also 6 References Area ( edit ) The total area of a circle is πr . The area of the sector can be obtained by multiplying the circle 's area by the ratio of the angle and 2π ( because the area of the sector is proportional to its angle , and 2π 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 ) \ cdot ( \ 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 πr by the ratio of L to the total perimeter 2πr . A = π r 2 ⋅ L 2 π r = r ⋅ L 2 ( \ displaystyle A = \ pi r ^ ( 2 ) \ cdot ( \ frac ( L ) ( 2 \ pi r ) ) = ( \ frac ( r \ cdot L ) ( 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 the central angle into degrees gives A = π r 2 ⋅ θ ∘ 360 ∘ ( \ displaystyle A = \ pi r ^ ( 2 ) \ cdot ( \ 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 + 2 r = θ r + 2 r = r ( θ + 2 ) ( \ displaystyle P = L + 2r = \ theta r + 2r = r ( \ theta + 2 ) ) where θ is in radians . Arc length ( edit ) Formula for Arc Length S = r θ ( \ displaystyle S = r \ theta ) where , S 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 : S = 2 π r θ 360 ( \ displaystyle S = 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 = 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 . See also ( edit ) 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 References ( edit ) Gerard , L.J.V. The Elements of Geometry , in Eight Books ; or , First Step in Applied Logic , London , Longman 's Green , Reader & Dyer , 1874 . p. 285 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Circular_sector&oldid=838806222 '' Categories : Circles Talk Contents About Wikipedia Azərbaycanca Беларуская Català Čeština ChiShona Dansk Deutsch Español Esperanto Euskara فارسی Français 한국어 Հայերեն Italiano Latviešu Lëtzebuergesch Lietuvių Magyar Nederlands 日本 語 Norsk Norsk nynorsk ភាសា ខ្មែរ Piemontèis Polski Português Русский Slovenščina کوردی Suomi Svenska தமிழ் ไทย Українська Tiếng Việt 吴语 中文 30 more Edit links This page was last edited on 29 April 2018 , at 11 : 52 ( UTC ) . About Wikipedia