Prisms and Cylinders

Excerpts

• Prisms and Cylinders

  *Two polyhedrons have the same volume if they can be made to coincide, or if they can be divided into parts which can be made to coincide in pairs*.

• Prisms and Cylinders

  Any prism, can be divided into triangular prisms by planes passing through one edge and each of the other non-adjacent edges.

• Prisms and Cylinders

  but no attempt has been made to *measure the space occupied* by such a solid. For this purpose we consider first a rectangular parallelopiped.

• Prisms and Cylinders

  Since the edge $BE$ is $5$ units long, $5$ such tiers will exactly fill the space within the solid. That is, $5\cdot 3\cdot 4 = 60$ is the number of cubic units in the solid.

• Prisms and Cylinders

  E.g., if the length is $5$ inches and the width is $\sqrt{3}$ inches, then the base cannot be exactly covered with equal cubes, however small.

• Prisms and Cylinders

  The form of statement in this theorem is the usual abbreviation for the more *precise* form:

• Prisms and Cylinders

  Indeed, in these cases approximate measurement only is possible, since no square unit exists in terms of which such areas can be *exactly* measured.

• Prisms and Cylinders

  *A cylinder has a definite lateral area and a definite volume which may be approximated as nearly as we please by taking the lateral areas and the volumes of successive inscribed or circumscribed prisms*.

• Prisms and Cylinders

  The lateral area of a cylinder can be computed only in case the perimeter of a right section can be found, and this is possible by elementary methods only in case the right section is a circle.

• Prisms and Cylinders

  The moving line is the *generator*, and the generator in any one of its positions is an *element* of the surface.

• Prisms and Cylinders

  In this case $r = r_1 = r_2$, and $e = h$, and the formulas of corollaries 1 and 2 become identical.

Equations

• Prisms and Cylinders

  \textit{Volume} = \textit{Length}\/ × \textit{Width}\/ × \textit{Height}.

  The volume of a rectangular parallelopiped with commensurable edges is the product of its length, width and height.

• Prisms and Cylinders

  \textbf{Volume} = \textbf{Length} × \textbf{Width} × \textbf{Height}.

  The volume of a rectangular parallelopiped with commensurable dimensions is given by the product of its length, width and height.

• Prisms and Cylinders

  V = l\cdot w\cdot h

  The volume of a rectangular parallelopiped is the product of its length, width and height.

• Prisms and Cylinders

  \dfrac{V}{V'} = \dfrac{a\cdot b\cdot c} {a'\cdot b'\cdot c'} = \dfrac{a}{a'}

  The volumes of two rectangular parallelopipeds are in the ratio of the products of their corresponding dimensions, and so in the ratio of a pair of dimensions when the other two are equal.

• Prisms and Cylinders

  S = 2 \pi re

  The lateral surface of a circular cylinder equals 2π times the radius of its right section times the length of an element.

• Prisms and Cylinders

  S = 2 \pi rh

  The lateral surface of a right circular cylinder equals 2π times its radius times its altitude.

• Prisms and Cylinders

  V = \pi r_1^2 e

  The volume of a circular cylinder equals π times the square of the radius of its right section times the length of an element.

• Prisms and Cylinders

  V = \pi r_2^2 h

  The volume of a cylinder with a circular base equals π times the square of the base radius times the altitude.

• Prisms and Cylinders

  V = \pi r^2h

  The volume of a right circular cylinder equals π times the square of its radius times its altitude.

Problems

No exercises in this chapter.