|Theorem 13.5||The lines containing the altitudes of a triangle are concurrent.|
|Theorem 13.6||Two medians of a triangle intersect at a point two-thirds of the distance from each vertex to the midpoint of the opposite side.|
|Theorem 13.7||The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.|
|Theorem 14.1||The lateral area of a right prism is the product of the perimeter of a base and the length of an altitude (L = ph).|
|Theorem 14.2||The lateral area of a right cylinder is the product of the circumference of a base and the length of an altitude. The lateral area of a right cylinder with radius r and altitude of length h is 2Prh.|
|Theorem 14.3||The total area of a right cylinder with radius of length r and altitude of length h is 2Pr^2 + 2Prh, or 2Pr(r + b).|
|Theorem 14.4||If the ratio of the lengths of corresponding edges of two similar polyhedra is then the ratio of the lateral areas and of the total areas is (a/b)^2.|
|Postulate 23||For any rectangular solid, the volume V = lwh, where 1, w, and b are the lengths of three edges with a common vertex.|
|Theorem 14.5||The volume of a cube with edges of length S is S3.|
|Postulate 24||If a solid is the union of two or more nonoverlapping solids, then its volume is the sum of the volumes of these nonoverlapping parts.|
Cavalieri's Principle: If two solids have equal heights, and if the cross sections formed by any plane parallel to the bases of both solids have equal areas, then the volumes of the solids are equal.
For any prism or cylinder, the volume is the product of the area of a base and the length of an altitude (V = Bh, where B = area of base and h = altitude).
|Corollary||The volume of a cylinder is Bh, or Pr^2h.|
|Theorem 14.7||The lateral area of a regular pyramid is one-half the product of the perimeter of the base and the slant height (L = 1/2pl).|
|Theorem 14.8||The lateral area L of a right cone is Prl. The total area (A) is, Prl + Pr^2 = Pr(l + r).|
|Theorem 14.9||The volume of a pyramid is one-third the volume of a prism with the same base and altitude as the pyramid. The volume of a cone is one-third the volume of a cylinder with the same base and altitude as the cone (V = 1/3Bh).|
|Corollary 1||The volume of a pyramid or cone is one-third the product of the area of its base and the length of its altitude (V = 1/3Bh).|
Note: The concepts in this collection may
not be entirely accurate.
They are for reference only.