 | Benjamin Greenleaf - Geometry - 1868 - 340 pages
...other as their bases ; and cylinders of equal bases are to each other as their altitudes. 577. Cor. 2. Similar cylinders are to each other as the cubes of...their bases. For the bases are as the squares of their radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each... | |
 | Benjamin Greenleaf - 1869 - 516 pages
...other as their bases ; and cylinders of equal bases are to each other as their altitudes. 577. Cor. 2. Similar cylinders are to each other as the cubes of...their bases. For the bases are as the squares of their radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each... | |
 | 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, the solidity of the cone... | |
 | 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... | |
 | 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... | |
 | Edward Olney - Geometry - 1872 - 472 pages
...second couplet of (1) by these equals, we have, PROPOSITION XТI. 409. Theorem. — The volumes of similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. DEM. — Using the same notation ns in the last demonstration, the student should... | |
 | 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"... | |
 | 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... | |
 | 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... | |
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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. 
