how do I find the arc length of a sector?

Question

Find the arc length of a sector with a central angle of 225 and a radius of 4.2 meters.

 

is it 

a) 4pi

b) 4pi/5

c) 5pi/5

d) 8pi    ?

 

Thank you for reading, I really appreciate your help.

 

Answer 1

Quick Answer: arc length = (central angle in radians) * radius = (225 degrees) * (pi radians) / (180 degrees) * (4.2 meters) = 5/4 * pi * (4.2 meters) = 5.25 * pi ... so none of those choices is correct.

 

Long Answer:

The definition of radian is the angle such that the arc of the sector created by that angle has a length of one radius. 

Since a circle's circumference is 2*pi*r, and one radian covers an arclength of r, it must take 2*pi radians to cover the full circumference. This is how we know that 2*(pi radians) = 360 degrees (or pi radians = 180 degrees). 

Radians are a very nice measure because they relate angles to arclengths in a simple way. From the definition, an angle of one radian corresponds to an arclength of one radius. Thus, an angle of say, 5 radians, corresponds to an arclength of 5 radii. 

From these definitions, I hope you can follow my quick answer above :)

Answer 2

Calculate circumference, C = 2(pi)r.

 

Take the appropriate proportion of it [θ/(2pi)]: 225/360•C      Note I used the ratio with degree measurements.

Answer 3

Use the formula

 

s = rθ

 

where:

s = arc length

r = radius

θ = angle