I’m not sure how useful it is to find the average radius of an ellipse, but it is interesting — and better with a picture!<\/p>\n
The distance from the center of a circle to any point on the circle is a constant — that’s the length of the radius.<\/p>\n The distance from the center of an ellipse to a point on the ellipse is not constant.\u00a0 The longest it can be is the semi-major axis, a<\/em>.\u00a0 The shortest it can me is the semi-minor axis, b<\/em>.\u00a0 But what do we mean by the average radius?\u00a0 One possible interpretation is that the average radius is the arithmetic mean of the two semi-axes.\u00a0 But here’s another interpretation:<\/p>\n What is the radius of the circle that would have the same area as the ellipse?<\/strong><\/p>\n Clearly, that radius would have to be more than b<\/em> and less than a<\/em>.\u00a0 So maybe it is their mean…but which kind of mean?<\/p>\n To answer this, you need to know two formulas:<\/p>\n A<\/span>circle<\/span><\/sub> = \u03c0r<\/em><\/span>2<\/span><\/sup><\/em><\/span><\/p>\n A<\/span>ellipse<\/span><\/sub> = \u03c0ab<\/em><\/span><\/span><\/p>\n So if the two areas are two be equal, we need r<\/em>2<\/sup><\/em> = ab.<\/em><\/p>\n In other words…<\/p>\n For a circle\u00a0to have the same area as an ellipse, the length of its radius must be the geometric mean of the lengths\u00a0of the ellipse’s semi-major and semi-minor axes!\u00a0<\/em><\/strong><\/p>\n I didn’t know that.\u00a0 And I was surprised to see a geometric mean pop up in class so shortly after my last post.\u00a0 Also, I should add: I don’t know how to show that the semi-major axis is the average orbit radius.\u00a0 If any of my fine students can come explain that to me, it is certainly worth a little extra credit.<\/p>\n","protected":false},"excerpt":{"rendered":" Here’s a quick\u00a0addition to\u00a0that last post on various types of means.\u00a0 By coincidence, yesterday,\u00a0a question\u00a0came up\u00a0in my AP physics class about the meaning of the average radius of an ellipse. We were looking at the laws that govern circular orbits.\u00a0 … Continue reading ![\"ellipse\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2015\/01\/ellipse.png\")
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