{"id":105,"date":"2014-04-20T14:31:52","date_gmt":"2014-04-20T18:31:52","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=105"},"modified":"2014-04-20T14:31:52","modified_gmt":"2014-04-20T18:31:52","slug":"angle-measurement-for-pizza-crust-lovers","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=105","title":{"rendered":"Angle Measurement for Pizza-Crust Lovers"},"content":{"rendered":"

Whether we measure a right angle to be 90 degrees or 100 grads, either way we are adopting an arbitrary scale.\u00a0 But when we say that its measure is \u03c0\/2 radians<\/strong><\/em>, we are using a system that is not arbitrary at all.\u00a0 It is based on a geometric comparison that works a lot like an aspect ratio.<\/p>\n

As we said earlier, the aspect ratio of a rectangle is the ratio of length to width.\u00a0 Intuitively, it is a measure of the \u201crectangularity\u201d.\u00a0 Long skinny rectangles have high aspect ratios.\u00a0 Squares have aspect ratio L\/W = 1.<\/p>\n

\"post8pic1\"<\/a><\/p>\n

One important thing to understand about aspect ratios: they don\u2019t tell you anything about the size of the rectangle.\u00a0 These two rectangles both have the same aspect ratio (2.5) though one is clearly bigger:<\/p>\n

\"post8pic2\"<\/a><\/p>\n

Aspect ratios tell you about the shape<\/i>, not the size.<\/p>\n

Also, as long as you measure the length and width with the same units, the aspect ratio is \u201cdimensionless\u201d.\u00a0 For a given rectangle, you get the same aspect ratio whether you measure your length and width in inches, feet, yards, miles\u2026it doesn\u2019t matter, as long as you use the same units for both measurements.<\/p>\n

\u00a0Onward to Angle Measurement<\/em><\/strong><\/p>\n

Here again are a collection of slices all cut from the same pizza but with different central angles:<\/p>\n

\"post7pic1\"<\/a><\/p>\n

And here\u2019s a critical fact to notice about these slices: for pizza of a given radius, the bigger the angle, the more crust you get.\u00a0 (Many of my mathematical insights involve food. Why is that?)<\/p>\n

The crust length is better known as \u201carc length\u201d.\u00a0 And now we can define a kind of aspect ratio.\u00a0 Instead of length-to-width, let\u2019s base this aspect ratio on the arc length and the radius.<\/p>\n

\"post8pic3\"<\/a><\/p>\n

The\u00a0arc length-to-radius\u00a0ratio is a natural and completely non-arbitrary way to measure the angles.\u00a0 All of the slices above have the same radius.\u00a0 So the biggest arc length-to-radius ratio is found on the right where the crust is longest.\u00a0 And that slice has the biggest central angle.<\/p>\n

For example, suppose you cut\u00a0two slices from a pizza with a radius of 12 inches, one with\u00a06 inches of crust and the other with 9 inches of crust.<\/p>\n

\u00a0\"post8pic4\"<\/a><\/span><\/strong><\/b><\/span><\/span><\/p>\n

\u00a0<\/b>We can define the measure of the central angle as the Angle Aspect Ratio<\/em>.<\/p>\n

For the smaller piece…<\/p>\n

Angle Aspect Ratio\u00a0= Arc Length \/ Radius = 6inches \/ 12 inches = 1\/2.<\/p>\n

And for the larger piece…<\/p>\n

Angle Aspect Ratio = Arc Length \/ Radius = 9inches \/ 12 inches = 3\/4.<\/p>\n

As expected, the larger Angle Aspect Ratio is associated with the larger central angle.\u00a0 You got more crust per radius with that angle.<\/p>\n

Also, just like aspect ratios for rectangles, it\u2019s the shape and not the size that matters. These two slices have the same angular aspect ratio.<\/p>\n

\"post8pic5\"<\/a><\/p>\n

The bigger slice has more crust but the crust-to-radius ratio is the same. So both of these angles have the same measure:<\/p>\n

\"post8pic6\"<\/a><\/p>\n

\u00a0<\/b>And yes, the real name for this system is called \u201cradian measure\u201d (and not “angle aspect ratio”, alas).<\/p>\n

In this system, the size of a right angle is not an arbitrary 90 degrees or 100 grads.\u00a0 Instead, it is the ratio of the arc length to the radius.\u00a0 But we can figure out that value.\u00a0 We know a full circle has a circumference of 2\u03c0r.\u00a0 So the arc length of a quarter circle must be 2\u03c0r \u00f7 4 =\u03c0r\/2.<\/p>\n

\"post8pic7\"<\/a><\/p>\n

Then, to find the radian measure, we\u00a0calculate the “angle aspect ratio” (as I stubbornly call it):<\/p>\n

\"post8pic8\"<\/a><\/p>\n

And now that we know that a right angle measures \u03c0\/2 radians, we can also say that<\/p>\n

\u03c0\u00a0radians = 180 degrees.<\/i><\/strong><\/p>\n

That\u2019s a useful fact if you want to convert back and forth from degrees to radians.\u00a0 But if all you know is that conversion factor, then you have missed the point. I know that some of my students think of radians the way they think of the metric system: another unfamiliar system you have to learn because math and science teachers insist on it.\u00a0 But seriously, mathematicians are as lazy as anyone else.\u00a0 They would never have adopted the system of radian measure if it didn\u2019t have advantages over the more commonly used degree measure.\u00a0 Some of these advantages don\u2019t show up until you are studying physics and calculus.\u00a0 But some of them can be seen sooner than that.\u00a0 Stay tuned\u2026<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Whether we measure a right angle to be 90 degrees or 100 grads, either way we are adopting an arbitrary scale.\u00a0 But when we say that its measure is \u03c0\/2 radians, we are using a system that is not arbitrary … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-1H","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/105"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=105"}],"version-history":[{"count":10,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/105\/revisions"}],"predecessor-version":[{"id":125,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/105\/revisions\/125"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=105"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}