 | William Chauvenet - Geometry - 1871 - 380 pages
...similar, we have and hence F v _ . r2 . h ~ r* ' h ~ h3 ~ r3 ' that is, similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. PROPOSITION VIII.— THEOREM. ^\ '. 1 \ 31. A frustum of any cone is equivalent to the sum of three... | |
 | William Chauvenet - Geometry - 1871 - 380 pages
...have H_R h ~ r' and F. v H h ' IP h3 that is, the volumes of similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii. THE CONE. 15. Definition. The area of the convex, or lateral, surface of a cone is called... | |
 | Elias Loomis - Geometry - 1871 - 302 pages
...as their bases ; cones of equal bases are to each other as their altitudes ; and similar cones are as the cubes of their altitudes, or as the cubes of the diameters of their bases. Cor. 2. If A represent the altitude of a cone, and R the radius of its base,... | |
 | Edward Olney - Geometry - 1872 - 472 pages
...the terms of the second couplet of (1) by these equals, we have, PROPOSITION XТI. 409. Theorem. — The volumes of similar cylinders are to each other...altitudes, or as the cubes of the radii of their bases. DEM. — Using the same notation ns in the last demonstration, the student should be able to give the... | |
 | William Chauvenet - Geometry - 1872 - 382 pages
...H_R ihr and hence v ~^.r 2 . h ~ r 2 ' h ~ h* " r' ' __ that is, similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. PROPOSITION VIII.— THEOREM. 31. A frustum of any cone is equivalent to the sum of three cones whose... | |
 | Edward Olney - 1872 - 270 pages
...^bh :: A" : a" : : H " : K\ QED PROPOSITION XII. 531, Theorem.—The volumes of similar cones are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. DEM. R and r being the radii of their bases, and H and h their altitudes, R" : r " :: H' : A' (?),... | |
 | Edward Olney - Geometry - 1872 - 562 pages
...: : A1 : a" : : H1 : h3. QED PROPOSITION XIL 831. Theorem. — Tlie volumes of similar cones are to each other as the cubes of their altitudes, or as the cubes of the radii of their DEM. R and r being the radii of their bases, and H and A their altitudes, R* : r* :: H« : A« (?),... | |
 | Charles Davies - Geometry - 1872 - 464 pages
...themselves are to each other as the cubes of their homologous edges. Cor. 2. Similar prisms are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines. PROPOSITION XX. THEOREM. Similar pyramids are to each other as the cubes... | |
 | William Chauvenet - Mathematics - 1872 - 382 pages
...similar, we have HR and xr'h H h ' IP h" that is, the volumes of similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii. THE CONE. 15. Definition. The area of the convex, or lateral, surface of a cone is called... | |
 | Adrien Marie Legendre - Geometry - 1874 - 500 pages
...pyramid S.AB CDE : pyramid S.abcde : : AB3 • ab" ; wluch was to be proved. Cor. Similar pyramids are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines. GENERAL FORMULAS. If we denote the volume of any prism by V, its base by... | |
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Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases. 
