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Before I continue my zealous advocacy for radian measure, I should acknowledge that other than Feynman, I have never seen anyone use radians while making an every-day casual reference to the size of an angle.\u00a0 Degree measure is more convenient here because it is more familiar.\u00a0 Familiarity is not a bad thing. Besides, even students who have completed a year of trigonometry and who have memorized their way around the unit circle (\u03c0\/6, \u03c0\/4, \u03c0\/3, \u03c0\/2\u2026) often struggle when I ask them to draw an angle that is approximately one radian.<\/p>\n
Try it yourself to see what I mean.\u00a0 Draw an angle of approximately one radian.\u00a0 But no fair converting!\u00a0 Though you could<\/em> use the fact that \u03c0 radians = 180 degrees to figure out what one radian is in degrees, that would evade the purpose of the question.\u00a0 The goal is to develop an intuitive feel for radian measure. So try to draw a radian without \u201cthinking in degrees\u201d.<\/p>\n <\/p>\n [Digression]<\/p>\n Early in the school year, students often ask \u201chow much is a kilogram\u201d?\u00a0 I know they want an answer that relates kilograms to pounds.\u00a0 But that kind of conversion factor doesn\u2019t help develop intuitive feel.\u00a0 I prefer to point out that one kilogram is the mass of a liter of water.\u00a0 And since all of my students walk around carrying half-liter bottles of water at all times (the hydration generation), they all know exactly what a half-kilogram feels like.<\/p>\n [\/digression]<\/p>\n <\/p>\n So how do we draw that one-radian angle?\u00a0 Let\u2019s rephrase that question in edible terms: draw a slice of pizza with crust-length equal to the radius!<\/p>\n Since the arc length is equal to the radius, it must be that the ratio is 1:1.\u00a0 That makes the central angle one radian.\u00a0 That\u2019s the angle Feynman was using to describe his errant throw.<\/p>\n Now let\u2019s get rid of the pizza and just look at the angle.\u00a0 Suppose you increase that angle just a little to pull the crust into a straight line (without changing its length).\u00a0 You would now have an equilateral triangle.<\/p>\n That tells us something interesting:\u00a0 one radian is a little less than the angle of an equilateral triangle.\u00a0 And since we already knew that the three of the angles of an equilateral triangle (in fact,of any triangle) add up to a straight line, it must take a little more than 3 radians to also add up to a straight line.<\/p>\n How much more?\u00a0 You already know that too:\u00a0 \u03c0 radians = 180 degrees. Right. But maybe now you have a little more of that radian feeling.<\/p>\n THIS JUST IN:<\/p>\n I found this on BusinessInsider.com in an article by Walter Hickey.<\/p>\n Nice.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \u201cI was never any good in sports.\u00a0 I was always terrified if a tennis\u00a0 ball would come over\u00a0 the fence\u00a0 and land near\u00a0 me, because I never could\u00a0 get it over the\u00a0 fence —\u00a0 it\u00a0 usually went about\u00a0 a\u00a0 radian … Continue reading ![\"post9pic1\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/04\/post9pic1.jpg\")
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