Circular sector

Types

A sector with the central angle of 180° is called a half-disk and is bounded by a diameter and a semicircle. QuadrantSectors 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 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 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 = \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 πr by the ratio of L to the total perimeter 2πr. 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}

Converting the central angle into degrees gives A = \pi r^2 \frac{\theta^\circ}{360^\circ}

Perimeter

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) where θ is in radians.

Arc length

The formula for the length of an arc is: 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 \pi r \frac{\theta}{360^\circ}

Chord length

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

Sources

References

1. Dewan, Rajesh K.. (2016). "Saraswati Mathematics". New Saraswati House India Pvt Ltd.
2. Uppal, Shveta. (2019). "Mathematics: Textbook for class X". [[National Council of Educational Research and Training]].
3. (2002). "Calculus I with Precalculus". [[Cengage.
4. Wicks, Alan. (2004). "Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus". Infinity Publishing.com.