When two rays intersect, we say that they form an angle.\u00a0 For example, the two rays could be cut along different radii of a pizza, intersecting at the center of the pie.\u00a0 Depending on how you make your cuts, you will get different sizes for your central angle.\u00a0 If you are hungrier, you cut a larger central angle.<\/p>\n
Since angles come in different sizes, we need to have a system for measuring them.\u00a0 And of course, we have a system that you have known forever.\u00a0 We call a full circle 360 degrees.\u00a0 We call a u-turn a \u201c180\u201d, short for 180 degrees.\u00a0 We call the central angle of a quarter of a pizza a 90-degree angle.\u00a0 That is also the measure of the angle you find in the corner of every rectangle.\u00a0 It\u2019s such a familiar system that you may not even realize that it is completely arbitrary.\u00a0 Who\u00a0chose 360 as\u00a0the\u00a0magic number for a full circle?\u00a0 (I believe it goes back to the\u00a0ancient Sumerians, but I haven’t even checked Wikipedia.)\u00a0 My point is that we didn\u2019t have to do it this way (and we don\u2019t always).<\/p>\n \u00a0<\/b><\/p>\n Another way we <\/strong><\/em>could<\/strong> measure angles<\/strong><\/em><\/p>\n I\u2019m just saying that this would work too:<\/p>\n Let\u2019s describe the size of an angle by comparing it to a right angle, defining the measure of each angle\u00a0as a\u00a0percent of the measure of a right angle.\u00a0 So for example, the angle made by the diagonal of a square is half the size of a right angle.\u00a0 You are used to calling that a 45 degree angle, but in our new system, that would be an angle measure of\u00a050 — the diagonal makes 50% of a right angle.<\/p>\n Here are some fun facts about our new system:<\/p>\n 1. A right angle has measure of 100. (It is 100% of a right angle!)<\/p>\n 2. A full\u00a0rotation has measure 400. (Cut a pizza into 4 quarters.\u00a0 The four right angles add up to a complete circle.<\/p>\n 3. The measures of the angles of a triangle still add up to the same as the measure of a straight line.\u00a0 They each have a\u00a0total\u00a0measure of 200 in this system.<\/p>\n 4. Equilateral triangles have 3 angles with the same measure: 200\/3 = 66.666\u2026<\/p>\n This seems arbitrary and unfamiliar \u2013 compared to our current system which is arbitrary <\/i>but at least familiar.\u00a0 But here\u2019s one other fun fact about our new system:<\/p>\n 5.\u00a0 It isn\u2019t new at all!\u00a0 It\u2019s an existing system.\u00a0 When you express an angle as a percent of a right angle, you are measuring the angle in \u201cgradians\u201d or \u201cgrads\u201d.\u00a0 This a real thing. It may even be one of the modes on your calculator (but no one ever uses it).<\/p>\n I\u2019ve read that gradians were used in navigation but I have never seen it done.\u00a0 To me, it feels like a quirk of history that we chose one arbitrary system over another.\u00a0 There is, however, another<\/em> way to measure the size of an angle that is not arbitrary at all.\u00a0 It is a natural way to describe different angles using ratios in a manner much like the way they are used to describe rectangles.\u00a0 This system has a real name: it\u2019s called \u201cradian measure\u201d (and your calculator has a mode for this as well).\u00a0 But we can also think of it as the \u201cAngular Aspect Ratio\u201d.\u00a0\u00a0 I\u2019ll describe how it works in my next post.<\/p>\n","protected":false},"excerpt":{"rendered":" If you have been following my recent posts, you may now be waiting for an impassioned defense of \u201ccompleting the square\u201d. \u00a0It\u2019s coming soon, with pictures (of course).\u00a0 But talking about the aspect ratio of a rectangle got me thinking … Continue reading ![\"post7pic1\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/04\/post7pic1.jpg\")
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