Pyramids and Cones

Excerpts

• Pyramids and Cones

  If a line through the fixed point moves so as to touch the boundary of the polygon and is made to traverse it completely, the line is said to generate a convex *pyramidal surface*.

• Pyramids and Cones

  The lateral area of a regular pyramid is equal to one half the product of its slant height and the perimeter of the base.

• Pyramids and Cones

  The volume of any pyramid is one third of the product of its base and altitude.

• Pyramids and Cones

  *A pyramid has a definite volume which is less than the combined volume of any set of circumscribed prisms and greater than that of any set of inscribed prisms*.

• Pyramids and Cones

  The volume of a frustum of a pyramid is equal to the combined volumes of three pyramids whose common altitude is the same as that of the frustum, and the areas of whose bases are those of the upper and lower bases of the frustum and the mean proportional between these areas.

• Pyramids and Cones

  This process may be repeated by doubling the number of planes drawn parallel to the base and thus doubling the number of inscribed or circumscribed prisms. By continuing in this way either set of prisms may be made to coincide as nearly as we please with the pyramid.

• Pyramids and Cones

  In this case every face is a triangle, and any one of them may be taken as the base.

• Pyramids and Cones

  by rotating a right triangle $PMB$ about one of its legs, $PM$, as an axis.

• Pyramids and Cones

  The circumscribed pyramids all have the same slant height as that of the cone, and in case of the inscribed pyramids, the slant height may be made to differ by as little as we please from that of the cone by making the number of faces great enough.

• Pyramids and Cones

  Evidently either of these processes may be repeated indefinitely and the surfaces of the inscribed or circumscribed pyramids may be made to lie as close to the surface of the cone as we please.

• Pyramids and Cones

  In the case of a cone which is not a right circular cone the slant height varies from point to point and the process of computation of § [unit:266.]266 fails.

• Pyramids and Cones

  The theorem of § [unit:272.]272 holds for any cone whatever, whether right or oblique.

• Pyramids and Cones

  His work on the circle, cone, cylinder, and sphere is reflected in our treatment of surfaces and volumes at the present day.

Equations

• Pyramids and Cones

  S = \tfrac{1}{2} l\cdot p

  The lateral area of a regular pyramid equals one half the product of its slant height and the perimeter of its base.

• Pyramids and Cones

  \dfrac{b'}{b} = \dfrac{\overline{PK'}^2}{\overline{PK}^2}

  When a pyramid is cut by a plane parallel to its base, the ratio of the section's area to the base's area equals the ratio of the squares of their distances from the vertex.

• Pyramids and Cones

  V = \tfrac{1}{3} bh

  The volume of any pyramid is one third of the product of its base area and its altitude.

• Pyramids and Cones

  V = \tfrac{1}{3} hb + \tfrac{1}{3} hb' + \tfrac{1}{3} h \sqrt{bb'} = \tfrac{1}{3} h [b + b' + \sqrt{bb'}]

  The volume of a frustum of a pyramid equals one third of its altitude times the sum of the two base areas and their mean proportional.

• Pyramids and Cones

  \dfrac{OF}{MB} = \dfrac{OP}{MP} = \dfrac{OG}{MC}

  In a plane section parallel to the base of a circular cone, the ratio of a perimeter point's distance from the section's center O to the base radius is the same as the ratio OP/MP, so equal base radii give equal distances from O.

• Pyramids and Cones

  S = \tfrac{1}{2}\cdot 2 \pi r\cdot l = \pi r l

  The lateral area of a right circular cone equals pi times the base radius times the slant height.

• Pyramids and Cones

  S = \tfrac{1}{2} (2\pi r + 2\pi r')l = \pi l(r + r')

  The lateral area of a frustum of a right circular cone equals pi times the slant height times the sum of the radii of its two bases.

• Pyramids and Cones

  V = \tfrac{1}{3}\cdot \pi r^2\cdot h = \tfrac{1}{3} \pi r^2 h

  The volume of a cone with a circular base is one third of pi times the square of the base radius times the altitude.

• Pyramids and Cones

  V = \tfrac{1}{3} h(b + b' + \sqrt{bb'})

  The volume of a frustum of a cone equals one third of its altitude times the sum of the two base areas and the square root of their product.

• Pyramids and Cones

  V = \tfrac{1}{3} \pi h(r^2 + {r'}^2 + rr')

  The volume of a frustum of a right circular cone equals one third of pi times its altitude times the sum of the squares of the two base radii and their product.

Problems

No exercises in this chapter.