Question

If r= 1 and 0 = 5π/4, what is the approximate arc length?

Answer 1

Answer:  Choice D

===============================================

Work Shown:

L = theta*r

L = (5pi/4)*1

L = 5pi/4

L = 3.92699081698724 approximately

L = 3.927 units

The formula above only works if theta is in radians.

If the angle is in degrees, then you need to convert using the rule

y = (pi/180)*x

where x is the angle in degrees and y is the equivalent angle measure in radians.

In the fourth step, I used my calculator to get that value. You could use pi = 3.14, but in this case, your teacher wants you to use the stored value of pi found in the calculator.

---------------------------

Extra info (optional section):

If you're curious why that formula works, then consider a circle with circumference C = 2*pi*r. This is the distance around the circle, aka the perimeter.

The arc length in red is only a fractional portion of the full perimeter. Specifically, that fractional amount is theta/2pi. We multiply this fraction by the full perimeter and simplify

L = arc length

L = (full circle circumference)*(fractional amount)

L = (2*pi*r)*(theta/2pi)

L = theta*r

The "2pi" terms cancel in the last step. Again, this formula only works if theta is in radians. This is due to the fractional portion theta/2pi.

There is an equivalent formula for theta being in degrees, which is

L = (theta/360)*2*pi*r

but this version is a bit messier in my opinion.