π and Radians
A representation of what is, its relation to the unit of angle measure called radians, and the difference between radians and degrees.
In this worksheet, the gridlines are separated by r units, where r is the radius of the two circles. d represents the distance the center of the circle has moved. As you change d by dragging the slider or by directly inputting the value at the bottom left, the blue segment gets wrapped around the circle. (Note: d also has units of r.)
1. is defined to be the ratio of the circumference of a circle to its diameter.
How do we know its value? Go through this problem to see one way of approximating it.
(A) When d=x, the distance the center of the circle has moved should be ___.
(B) When d=x, the length of the portion wrapped around the circle should be ___. Check your answer with the number in blue.
(C) What are (A) and (B) in terms of the diameter? Hint: d is in units of radiuses and radius = 1/2*diameter.
(D) When , how much of the circle appears to be wrapped?
(E) Remember d=3.141592 means the center has moved 3.141592 radiuses over. What is (D) in terms of the diameter?
(F) Repeat (D) and (E) for .
(G) = (circumference)/(diameter) = # of times a diameter can be wrapped around its circle. So based on (F), ___.
If you haven't already done so, slide d from 0 all the way to to see what you just did, without interruptions. Note that . Instead, . is actually an irrational number.
2. Radians are a unit of measure defined to be the # of radiuses that go into the arc subtending the .
This makes sense since the angle is completely determined by how long the arc is compared to its radius.
(A) When d=1, the arc wrapped around the circle has length ___. Check your answer with the number in blue.
(B) How many radiuses can be wrapped around the arc from (A)? Hint: d is in units of radiuses.
(C) subtended by the arc from (A) (i.e. the green ) should be ___ radians. Check your answer with the value in green.
(D) Repeat (A)-(C) for d=x.
Notice that arc length and radius both have units of length. Since radians = (arc length)/(radius), you can see that radians are dimensionless (i.e. they are pure numbers). Also notice that radians = 360. So to convert from radians to degrees, multiply by .
Radians are a much more natural unit of measure than degrees, and therefore, most of mathematics uses radians instead of degrees. However, in this course, you will not be using radians. All measures will be given in degrees. Therefore, when doing trigonometry problems, MAKE SURE YOUR CALCULATOR IS SET TO DEGREES, NOT RADIANS. Typically, you can see which mode you're in by looking for a "deg" symbol or a "rad" symbol on the screen. If you're in the wrong mode, change it, or else your answers for trigonometry problems will likely come out all wrong.