The formula for the perimeter of a circle is 2πr. But the formula for the perimeter of a sector is not that straightforward. The sector is part of a circle or you may call it a slice as you see in a pizza. The perimeter of a sector in a circle is dependent on the angle that it makes at the center. The perimeter of a sector is called the circumference of the sector. In this article, we discuss the formula for the perimeter of a sector in a circle.
Formula for the perimeter of a sector
Let us derive the formula for the perimeter 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. We can clearly see that the perimeter will be the sum of radius + radius + arc length = 2r + arc length.
Let us now calculate arc length:
For a circle of angle 360o we know that the perimeter is => 2πr
For a circle of angle θ the arc length let us assume as => x
using unitary method x = (θ/360)*2πr ———–(1)
So the formula for the perimeter of sector will be: 2r + (θ/360)*2πr = 2r[1 + (θ*π)/180]
What is the maximum possible perimeter of a circle?
The formula for the perimeter of a sector is 2r[1 + (θ*π)/180]. The maximum possible value θ can reach is 360o. But at θ = 360, there will be no sector, it will be a complete circle. So θ can attain any value less than 360. At 360 the value of perimeter of the sector will become 2r + 2πr. So, the value of the perimeter of a sector is always < (2r + 2πr).
What is the least possible perimeter of a sector?
The least possible value of θ is 0o. But at θ = 0, there will be no sector, it will be a complete circle. So θ can attain any value greater than 0. At 0 the value of the perimeter of the sector will be 2r. So, the value of the perimeter of a sector is always >2r
The range of value of the perimeter of a sector: 2r<perimeter>(2r + 2πr).
Units for the perimeter of a sector in a circle
The units for the perimeter will be always in units like centimeters, meters, feet, kilometers, etc. But the unit of area of a sector will be in square units like m2, cm2, km2, etc.
Calculator for Sector perimeter
You can use this calculator for the calculation of the perimeter of a sector.
You can use this calculator for the perimeter of a sector based on sector angle and radius. You can also calculate sector arc length and sector area using this tool.
Numericals:
A circle with a radius of 10 m has a sector making an angle of 60° at the center. Find the perimeter of the sector. (π = 3.14)
Given values => radius = 10 m; angle of sector at center = 60°
Formula of perimeter of sector = 2r[1 + (θ*π)/180]
Thus perimeter = 20[1+ (60*3.14)/180] = 40.92 m
The arc length of a sector in a circle is 40 cm. The area of the complete circle is 628 cm2. Find the perimeter of the sector.
Area of sector = πr2 = 628
r = 4.47 cm
Perimeter of sector = 2*radius + arc length = 2*4.47 + 40 = 48.94 cm
The area of a circle is 628 cm2. A sector in the circle forms an angle of 60° st in the center of the circle. Find the arc length of the sector.
Area of circle = πr2 = 628 which implies r = 4.47 cm
Formula for perimeter of a sector = 2r[1 + (θ*π)/180]
perimeter = 2*4.47[1+ (60*3.14)/180] = 18.2972
Perimeter = 2*radius + arc length
18.2972 = 2*4.47 + arc length
arc length = 18.2972 – 8.94 = 9.35 cm
See Also
Area of a sector formula
Perimeter of right angled triangle
Every square is a rectangle
Orthocenter of a triangle
Volume of a parallelepiped
