One of the first geometric formulas we learn in plane\u00a0geometry\u00a0is that the area of a triangle is:<\/p>\n\n\n\n
Area of a Triangle = (1\/2) * Base Width * Height.<\/p>\n\n\n\n
So it is natural to wonder how this might generalize to pyramids in n-dimensional geometry. For instance, in 3-dimensions, the volume of a pyramid is:<\/p>\n\n\n\n
Volume of Pyramid = (1\/3) * Base Area * Height.<\/p>\n\n\n\n
The same formula actually holds for a cone in 3-dimensions as well. Traditionally, one thinks of a cone as an object whose base B is circular, but in fact when the base is any shape, mathematicians still call the object a\u00a0cone over B<\/em>, and the formula above still holds for a 3-dimensional cone over any shape B. In general, the\u00a0cone<\/em>\u00a0over any n-dimensional object B is the (n+1)-dimensional object formed by taking a point P outside the n-dimensional hyperplane spanned by B and taking the union of all the line segments from P to points in B. And the volume of such a cone is:<\/p>\n\n\n\n Volume of a Cone over B = (1\/n+1) * Volume of B * Height.<\/p>\n\n\n\n Here, the “Height” is the distance from P from the hyperplane spanned by B.<\/p>\n\n\n\n Presentation\u00a0Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> (Volume of B)*(x\/H)n<\/sup>\u00a0dx,<\/p>\n\n\n\n and integrating this from x=0 to x=H yields the formula above. Moreover, we can see that the factor (1\/n+1) emerges from integrating the xn<\/sup>\u00a0in the expression above!<\/p>\n\n\n\n How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> One of the first geometric formulas we learn in plane\u00a0geometry\u00a0is that the area of a triangle is: Area of a...<\/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],"class_list":["post-393","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-geometry","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/393","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=393"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/393\/revisions"}],"predecessor-version":[{"id":1698,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/393\/revisions\/1698"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=393"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Although, the concept of volume in\u00a0n-dimensional\u00a0space is something that students sometimes find difficult to comprehend, one may motivate the idea by explaining that the notion of volume is basically a way to quantify the “size” of a set in n-dimensional space in a way that is translation-invariant.<\/p>\n\n\n\n
The factor of (1\/n+1) is probably the most interesting part about this formula. One way to see where this comes from is to use\u00a0calculus. Consider a thin slice of the Cone over B, cut by planes parallel to the base B. This slice has cross-sectional volume that is a similar figure to B, except that in each dimension it has been scaled by (x\/H). So, if the thickness of the slice is represented by dx, the volume of this slice is represented by:\u00a0<\/p>\n\n\n\n
Su, Francis E., et al. “Volume of a Cone in N Dimensions.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"