Radian and Degree Measure

Question

Find the arc-length of a circle with the given radius r and central angle θ. Give the answer in the given unit of measure, rounded to the nearest hundredth.

r = 46 in.; θ = 315°

_____ in.

Answer 1

Hi Niki,

The first thing we need in our toolbox for this problem is the equation for determining arc length.

where s is the arc length

r is the radius of the circle

θ is the interior angle in radians

So now that we have that equation how can we fit the given values for r and θ into it? You'll notice that θ must be in radians, but the problem gives it to us in degrees. The first step is to convert 315º to radians.

1 radian = 180º/π

1 degree = π/180º

A great way to think about unit conversions is to ask "how can we get rid of the degree unit?" And unit/unit is just unitless. So to turn 315º into radians we have to use the equation that will cancel out the degree and leave only the radian (the unit attached to the π).

315º × (π/180º) = (315º / 180º) × π = 7π/4 rad

Now we can use our arc length equation correctly.

s = rθ

s = 46 in × 7π/4 = 322π/4 ≈ 252.8982086 in (using a calculator)

And rounded to the nearest hundredth we get 252.90 in

Answer 2

s = radian x radius = 46 x 315pi/180 = 46pi(35/20) = 46pi(7/4) = 23pi(7/2)

= 161pi/2

= 80.5pi inches

= about 80.5(3.14)

= about 252.90 inches