Octant (plane geometry)

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

A circular sector, also known as circle sector or disk sector or simply a sector (symbol: ⌔), is the portion of a disk (a closed 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.

Types

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.

Area

The total area of a circle is πr2. The area of a sector in terms of L can be obtained by multiplying the total area πr2 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}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}} {\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{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 2π (because the area of the sector is directly 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}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}} {\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}} Another approach is to consider this area as the result of the following integral: A = ∫ 0 θ ∫ 0 r d A = ∫ 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}dA=\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}}} {\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dA=\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[2] A = π r 2 θ ∘ 360 ∘ {\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}} {\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}

Arc length

The formula for the length of an arc is:[3] L = r θ {\displaystyle L=r\theta } {\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.[4]

If the value of angle is given in degrees, then we can also use the following formula by:[5] L = 2 π r θ 360 ∘ {\displaystyle L=2\pi r{\frac {\theta }{360^{\circ }}}} {\displaystyle L=2\pi r{\frac {\theta }{360^{\circ }}}}

Perimeter

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)} {\displaystyle P=L+2r=\theta r+2r=r(\theta +2)} where θ is in radians.

Chord length

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}}} {\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

References

  1. Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-8173358371.
  2. 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.
  3. 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.
  4. 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.
  5. Uppal (2019).

Sources