A sector is a part of a circle that comprises an arc with two radii like a pizza slice. The formula for the area of a circle is well known as πr2. The area of a sector is dependent on the angle that it makes at the center. In this article, we discuss the formula along with the numerical value for the area of a sector in a circle.
The formula for the area of a sector
Let us derive the formula for the area of a circle with radius r.
In the above image, you can see a circle with a radius r and a sector angle of θ.
For a circle of angle 360o, we know that the formula for area is => πr2
For a circle of angle θ let us assume the area as => x
using unitary method x = (θ/360)*Ï€r2 ———–(1)
So the formula for the area of a sector will be: (θ/360)*πr2
What is the maximum possible area of a circle?
The formula for the area of a sector is (θ/360)*πr2. The maximum possible value θ can reach is 360o, but that will mean a complete circle without any sector. So θ can attain any value less than 360. So, the value of the area of a sector is always less than πr2.
What is the least possible area of a sector?
The least possible value of θ is 0o. But at θ = 0, there will be no sector. So θ can attain any value greater than 0. At 0 the value of the area of the sector will be 0. So, the value of the area of a sector is always >0
The range of value of the area of a sector: 0<area>Ï€r2.
Units for the area of a sector in a circle
The unit of area of a sector will be in square units like m2, cm2, km2, etc.
Calculator for an area of a sector
You can use this calculator for the calculation of area, arc length, and chord length based on entries for radius and angle at the center.
Numericals:
A circle with a radius of 100 cm has a sector making an angle of 60° at the center. Find the area of the sector. (π = 3.14)
Given values => radius = 100 cm; angle of sector at center = 60°
Formula for area of a sector = (θ/360)*πr2
Thus area = 100*100*3.14[(60)/180] = 5,233.3 cm2
A sector has 1/5th the area of a circle with a radius of 5 cm. Find the angle at the center of the sector.
Area of circle = πr2 = 78.5 cm2
area of sector = 1/5(area of sector) = 78.5/5 = 15.7 cm
Formula for area of sector = (θ/360)*πr2
15.7 = (θ/360)*πr2
θ = (15.7 * 360)/78.5 = 72o
A circle has a radius of 5 m. Is a sector with an area of 80 m2 possible inside the circle. Justify your answer.
The maximum value of the area that a sector can have is slightly less than πr2.
Ï€r2 = 3.14*5*5 = 78.5 m2
This value of 80 exceeds the area of the circle at 78.5. So such a sector with an area of 80 m2 is not possible for a circle of radius of 5m.
See Also
Perimeter of a sector-Formula
Perimeter of right angled triangle
Every square is a rectangle
Orthocenter of a triangle
Volume of a parallelepiped
