{"id":373,"date":"2019-06-26T23:48:04","date_gmt":"2019-06-26T23:48:04","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=373"},"modified":"2019-11-18T21:43:09","modified_gmt":"2019-11-18T21:43:09","slug":"arclength-surprise","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/arclength-surprise\/","title":{"rendered":"Arclength Surprise"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Consider a\u00a0unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is placed, the area between S and the x-axis plus the area between S and y-axis is constant! Moreover, that constant is equal to the length of S:<\/p>\n\n\n\n

A + B = s2<\/sub> – s1<\/sub>.<\/p>\n\n\n\n

In Figure 1, note that regions A and B overlap; in that portion the area is counted twice. The quantity (s2<\/sub> – s1<\/sub>) represents the length of S along the arc from s2<\/sub> to s1<\/sub>. <\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw a couple of pictures with arcs of the same length in different positions. Perhaps assign the computation of the areas as a fun homework exercise.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
Use\u00a0calculus, or sector-triangle formulas from\u00a0geometry, to compute the corresponding areas.<\/p>\n\n\n\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Arclength Surprise.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References<\/strong>:
Putnam examination, 1998, in The William Lowell Putnam Mathematical Competition 1985-2000.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider a\u00a0unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[20,7,8,4,75],"class_list":["post-373","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-geometry","tag-medium","tag-unit-circle"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/373","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=373"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/373\/revisions"}],"predecessor-version":[{"id":1345,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/373\/revisions\/1345"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=373"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}