Did you know there is a version of the\u00a0Pythagorean Theorem\u00a0for right triangles on spheres?<\/p>\n\n\n\n
First, let’s define precisely what we mean by a spherical triangle. A great circle<\/em> on a sphere is any circle whose center coincides with the center of the sphere. A spherical triangle<\/em> is any 3-sided region enclosed by sides that are arcs of great circles. If one of the corner angles is a right angle, the triangle is a spherical right triangle<\/em>.<\/p>\n\n\n\n In such a triangle, let C denote the length of the side opposite right angle. Let A and B denote the lengths of the other two sides. Let R denote the radius of the sphere. Then the following particularly nice formula holds:\u00a0<\/p>\n\n\n\n cos(C\/R) = cos(A\/R) cos(B\/R).\u00a0<\/p>\n\n\n\n Presentation\u00a0Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> C2<\/sup>\u00a0= A2<\/sup>\u00a0+ B2<\/sup><\/p>\n\n\n\n as R goes to infinity! This should make sense, since as R goes to infinity, spherical geometry becomes more and more like regular planar\u00a0geometry!<\/p>\n\n\n\n By the way, there is a “hyperbolic geometry” version, too. Can you guess what it says? See the reference.<\/p>\n\n\n\n How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by:<\/strong> Did you know there is a version of the\u00a0Pythagorean Theorem\u00a0for right triangles on spheres? First, let’s define precisely what we...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[8,4,12,44,163],"class_list":["post-444","page","type-page","status-publish","hentry","tag-geometry","tag-medium","tag-other","tag-sphere","tag-triangles"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/444","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=444"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/444\/revisions"}],"predecessor-version":[{"id":1650,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/444\/revisions\/1650"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=444"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Verify the formula is true in some simple examples: such a triangle with two right angles formed by the equator and two longitudes. For more on spherical triangles, see the Fun Fact on\u00a0Spherical Geometry.<\/p>\n\n\n\n
This formula is called the “Spherical Pythagorean Theorem” because the regular Pythagorean theorem can be obtained as a special case: as R goes to infinity, expanding the cosines using their\u00a0Taylor series\u00a0and manipulating the resulting expression will yield:\u00a0<\/p>\n\n\n\n
Su, Francis E., et al. “Spherical Pythagorean Theorem.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
D. Velian, “The 2500-Year-Old Pythagorean Theorem”, Math. Mag.<\/em>, 73(2000), 259-272.<\/p>\n\n\n\n
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"