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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. "
Mathematical Dictionary and Cyclopedia of Mathematical Science Comprising ... - Page 146
by Charles Davies, William Guy Peck - 1859 - 592 pages
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A Treatise on Elementary Geometry: With Appendices Containing a Collection ...

William Chauvenet - Geometry - 1871 - 368 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...
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A Treatise on Elementary Geometry: With Appendices Containing a Collection ...

William Chauvenet - Geometry - 1871 - 368 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...
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Elements of Geometry, Conic Sections, and Plane Trigonometry

Elias Loomis - Geometry - 1871 - 58 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,...
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A Treatise on Special, Or Elementary Geometry

Edward Olney - Geometry - 1872 - 239 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...
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A Treatise on Elementary Geometry: With Appendices Containing a Collection ...

William Chauvenet - Geometry - 1872 - 368 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...
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A Treatise on Special Or Elementary Geometry

Edward Olney - 1872 - 201 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' (?),...
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A Treatise on Special Or Elementary Geometry, Volumes 1-2

Edward Olney - Geometry - 1872
...: : 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 (?),...
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Elements of Geometry and Trigonometry from the Works of A.M. Legendre ...

Charles Davies - Geometry - 1872 - 455 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...
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A Treatise on Elementary Geometry: With Appendices Containing a Collection ...

William Chauvenet - Geometry - 1872 - 368 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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Elements of Geometry and Trigonometry: From the Works of A.M. Legendre

Adrien Marie Legendre - Geometry - 1874 - 455 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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