There are some geometrical shapes that are considered as the building block of geometry. One of those important geometrical shapes is a circle. A circle can be divided into different portions, such as if you divide the circle into half, it will become a semi-circle, however, if you divide a circle into four portions then each portion will be called a quarter circle. What if we are interested in something that is neither of the above? Consider the below picture.
This grey region is neither a semi-circle nor a quarter circle, then what it is? We call it a sector. A circular sector is the portion of a circle bounded by two radii. A sector doesn't have a perfect shape, if the region in a circle isn't a semi-circle nor a quarter circle then it is a sector. Sectors are very important in the geometry world and that is why we dedicated these resources for sectors.
Circular Sector Area
Finding the area of a circle is easy and finding the area of semi-circle and quarter circle is also easy. However, finding the area of a sector is a bit complex. A circle has 
, to find the area of a sector, you need to find the ratio of angles of sector and circle and then multiply it with the circle area. The ratio of angles can be found by dividing the sector's angle by .
Sometimes, angle can be in radian units. Therefore, replace by 
.
Arc Length
The same method is been applied to arc length too but with a small change. Find the ratio of angles and then multiply it by the circumference of the circle.
In case you are working in the radian units then replace by .
Perimeter of the Sector
This is one of the most frequently asked questions, can we call arc length the perimeter of the sector? Absolutely not! You can't even assume it! Arc length finds you the circumference of the sector. A sector contains radii and arc length. To find the perimeter of the sector, you need to add the radius too.
The above formula is for arc length, let's find the perimeter of the sector.
In the case of radian units:
Examples
Find the area of a circular sector whose chord is the side of the square inscribed in a circle with a 
radius.
The area of a circular sector of 
is 
cm. Find the radius and the length of the circle to which it belongs.
Of course, radius cannot be a negative entity.
Calculate the area of a circular sector whose chord is the side of an inscribed equilateral triangle in a circle with a 
radius.
The rope that attaches a swing to a tree is 
long and the maximum difference between trajectories is an angle of 
. Calculate the maximum distance travelled by the seat of the swing when the swing angle is described as the maximum.
A lighthouse sweeps its light across the horizon of the ocean at an angle of amplitude of 
degrees. If the maximum range of the beacon is 
miles, what is the maximum length of the corresponding arc?
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I’m just curious if the area between the polygon and the circumscribed circle has a name.
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html