Elements of geometry and trigonometry . se SA : Sa : :SX : Sx (Book VI. Prop. XV.) ; hence ABCDE : abcde : :XYZ : xyz ; hence the sections abode, xyz, are to each otheras the bases ABCDE, XYZ. Cor. 2. If the bases ABCDE, XYZ, are equivalent, any sec-tions abcde, xyz, made at equal distances from the bases, willbe equivalent likewise. PROPOSITION IV. THEOREM. The convex surface of a regular pyramid is equal to the périmeter of its base rnultipUed by half the slant height. For, since the pyramid is legular, tliepointO,in which the axis meets the base,is the centre of the polygon ABCDE(Def. 14.)
Elements of geometry and trigonometry . se SA : Sa : :SX : Sx (Book VI. Prop. XV.) ; hence ABCDE : abcde : :XYZ : xyz ; hence the sections abode, xyz, are to each otheras the bases ABCDE, XYZ. Cor. 2. If the bases ABCDE, XYZ, are equivalent, any sec-tions abcde, xyz, made at equal distances from the bases, willbe equivalent likewise. PROPOSITION IV. THEOREM. The convex surface of a regular pyramid is equal to the périmeter of its base rnultipUed by half the slant height. For, since the pyramid is legular, tliepointO,in which the axis meets the base,is the centre of the polygon ABCDE(Def. 14.) ; hence the lines OA, OB, OC,&c. drawn to the vertices of the base,are equal. In the right angled triangles S AO,SBO,Uie bases and perpendiculars are equal :À-biice the hypothenuses are equal : andKmay be proved in the same way that•II the sides of the right pyramid areequal. The triangles, therefore, whichform the convex surface of the prism are9\\ equal to each other. But the area ofiHiher of these triangles, as E8A, is equal. BOOK VU. 147 to its base EA multiplied by half the perpendicular SF, whichis the slant heiuiit of the pyramid : hence the area of all the tri-angles, or the convex surface of the pyramid, is equal to theperimeter of the hase multiplied by half the slant height. Coi\ The convex surface of the friislum of a regular pyra-mid is equal to halfUie perimeters of lis upper and lower basesmultiplied by its slant height. For, since the section alKde is similar to the base (Prop. III.),«md since the base ABODE is a regular polygon (Def. 14.), itfollows that tlie sides ea^ ah, be, cd and de are all equal to eachother. Hence Ûie convex s(irf*ce of the frustum ABCDE-û?is formed by the equal trapcsoids E\a, &:c, and theperj>endicular distance between the parallel sides of either ofthese trapezoids is equal to Ff tlic slant height of the the area of either of the trapezoids, as AEcm, is equal to*(EA4-crf) X Ff (Book IV. Prop. VII.) : licnce die ar
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