Angles in a unit circle word problem

Question

The ferris wheel as shown in the figure has 12 equal parts and one unit in radius moving clockwise direction. Suppose you are at seat #1, what angle in radian measure you will take in order to reach in the position of seat #6? What about at seat #11?

Answer 1

The important part of your picture, the angles from seat 1 to seat 6 and seat 1 to seat 11, probably looks something like this (not to scale):

s11 = 60°

/

/

/________ s1 = 0°

⁄

⁄

⁄

s6 = 210°

Note that the problem states seats 1-12 are ordered clockwise, but the angles you listed in your comment are measured counter-clockwise, so as the seat number gets bigger, the angle actually gets smaller. I don't know if that was intended by the question, but it doesn't matter much anyway.

First thing you should do is measure the smallest angle to get from seat 1 to each other seat. For seat 11, it's intuitively 60° as you listed, but is 210° the smallest angle to seat 6? I think not! 210° is what we get when we go counter-clockwise around the circle, but if we travel clockwise instead, it's a shorter trip. We can measure this trip by starting, not at 0°, but 360° (which is the same position as 0°, and going backwards until we reach 210°. To get that distance, what's the difference between 360° and 210°? (150°!)

Now that we know the angle measures in degrees, we have to turn them into radians. A radian measures how many radii it takes to travel around the whole circle. If that sounds confusing, don't worry, all you really need to know is that an angle of 2π wraps around the circle once; a.k.a., 2π radians is the same as 360°.

Notice, this sets up a proportion: if we know what fraction of 360° are angle is, we can apply that (by multiplying) to 2π to get our answer in radians. Here's how it looks for each angle:

s11 60° 1 1 π

———— = ———— = — -> so s11 is 1/6 of the total. In radians: 2π X — = —

360° 360° 6 6 3

s6 150° 5 5 5π

———— = ———— = —— -> so s6 is 5/12 of the total. In radians: 2π X —— = ——

360° 360° 12 12 6

So the angles for s11 and s6 are π/3 and 5π/6 respectively. You can combine these steps into one equation: θ = 2π * (θ° / 360°) which simplifies to θ = θ° * π/180°, which may be a formula that you've come across before. The reverse (radians to degrees) can be done as well: θ° = θ * 180 / π. Notice the two different thetas, θ and θ°. The first is the angle in radians and the second is angle in degrees.

Hope that makes sense, let me know if you need further clarification :)