SIMILAR SOLIDS

Excerpts

• SIMILAR SOLIDS

  The fact that the ratio of the areas of corresponding surfaces of similar solids is equal to the *square* of their ratio of similitude, while the ratio of their volumes equals the *cube* of this ratio is one of the most important and far-reaching conclusions of geometry.

• SIMILAR SOLIDS

  Any two figures which have a center of similitude are similar.

• SIMILAR SOLIDS

  This proposition may be rendered evident by noticing that any two similar three-dimensional figures may be built up to any degree of approximation by means of pairs of similar tetrahedrons similarly placed.

• SIMILAR SOLIDS

  The essential property of all such contrivances is that one point $O$ is kept fixed, while two points $A$ and $B$ are allowed to move so that $O$, $A$, and $B$ always remain in a straight line, and so that the ratio $OA : OB$ remains the same.

• SIMILAR SOLIDS

  Note that the ratio of similitude of two similar figures may be obtained from the ratio of any pair of their corresponding linear dimensions.

• SIMILAR SOLIDS

  Thus the ratio of the weights of two similar shells used in gunnery, or the ratio of the weights of two men of similar build, may be found when their ratio of similitude is known.

Equations

• This equation is in The Sphere (The Sphere)

• SIMILAR SOLIDS

  S = 2\pi r(r + h)

  The total area of a right circular cylinder equals 2 pi times the radius times the sum of the radius and the altitude.

• This equation is in Prisms and Cylinders (Prisms and Cylinders)

• SIMILAR SOLIDS

  V = \pi {r'}^2h'

  The volume of the second right circular cylinder equals pi times the square of its radius times its altitude.

• SIMILAR SOLIDS

  \dfrac{r}{r'} = \dfrac{h}{h'}

  For two similar cylinders the ratio of radii equals the ratio of altitudes.

• SIMILAR SOLIDS

  \dfrac{s}{s'} = \dfrac{S}{S'} = \dfrac{r^2}{r'^2} = \dfrac{h^2}{h'^2}

  For two similar right circular cones, the ratios of lateral areas and total areas equal the squares of the ratios of radii and altitudes.

• SIMILAR SOLIDS

  \dfrac{s}{s'} = \dfrac{S}{S'} = \dfrac{r^2}{r'^2} = \dfrac{h^2}{h'^2} \text{ and } \dfrac{V}{V'} = \dfrac{r^3}{r'^3} = \dfrac{h^3}{h'^3}

  For two similar right circular cones, the area ratios equal the squares of the radius and altitude ratios, and the volume ratio equals their cubes.

• SIMILAR SOLIDS

  \dfrac{V}{V'} = \dfrac{PA \cdot PB \cdot PC}{P'A' \cdot P'B' \cdot P'C'}

  Two tetrahedrons with an equal trihedral angle at P have volumes in the ratio of the products of the edges meeting at that vertex.

• SIMILAR SOLIDS

  \dfrac{V}{V'} = \dfrac{\overline{PA}^3}{\overline{P'A'}^3}

  The volumes of two similar tetrahedrons are in the ratio of the cubes of corresponding edges.

• SIMILAR SOLIDS

  OA : OA' = OB : OB'

  Two figures have a center of similitude O when the segments from O to corresponding points stand in a constant ratio.

• SIMILAR SOLIDS

  \dfrac{V}{V'} = \dfrac{\overline{AB}^3}{\overline{A'B'}^3}

  The volumes of any two similar polyhedrons are proportional to the cubes of their corresponding edges.

• SIMILAR SOLIDS

  \dfrac{\text{area } ABC}{\text{area } A'B'C'} = \dfrac{m^2}{n^2}

  Corresponding triangles of two similar figures have areas in the square of their ratio of similitude.

• SIMILAR SOLIDS

  \dfrac{\text{vol.\ } ABCD}{\text{vol.\ } A'B'C'D'} = \dfrac{m^3}{n^3}

  Corresponding tetrahedrons of two similar figures have volumes in the cube of their ratio of similitude.

• SIMILAR SOLIDS

  w = kh^3

  Assuming schoolboys' weights vary as the cube of their heights, weight equals a constant k times height cubed.

Problems

No exercises in this chapter.