Introduction
Let’s go from two dimensions to three. Here are some formulas for surface areas and volumes of common geometric solids. The threespace involved is flat ; that is, it obeys the laws of euclidean geometry. These formulas hold in newtonian physics (although in relativistic physics they may not).
Volume Of Pyramid
Suppose that we have a pyramid whose base is a polygon with area A and whose height is h (Fig. 435). The volume V of the pyramid is given by
V = Ah /3
Surface Area Of Cone
Suppose that we have a cone whose base is a circle. Let P be the apex of the cone, and let Q be the center of the base (Fig. 436). Suppose that line segment PQ is perpendicular to the base so that the object is a right circular cone . Let r be the radius of the base, let h be the height of the cone (the length of line segment PQ ), and let s be the slant height of the cone as measured from any point on the edge of the circle to the apex P . Then the surface area S of the cone (including the base) is given by either of the following formulas:
S = π r ^{2} + π rs
S = π r ^{2} + π r ( r ^{2} + h ^{2} ) ^{1/2}
The surface area T of the cone (not including the base) is given by either of the following:
T = π rs
T = π r ( r ^{2} + h ^{2} ) ^{1/2}
Fig. 436 Surface area of a right circular cone.
Volume Of Conical Solid
Suppose that we have a cone whose base is any enclosed plane curve. Let A be the interior area of the base of the cone. Let P be the apex of the cone, and let Q be a point in the plane X containing the base such that line segment PQ is perpendicular to X (Fig. 437). Let h be the height of the cone (the length of line segment PQ ). Then the volume V of the corresponding conical solid is given by
V = Ah /3
Surface Area Of Right Circular Cylinder
Suppose that we have a cylinder whose base is a circle. Let P be the center of the top of the cylinder, and let Q be the center of the base (Fig. 438). Suppose that line segment PQ is perpendicular to both the top and the base so that we have a right circular cylinder . Let r be the radius of the cylinder, and let h be the height (the length of line segment PQ ). Then the surface area S of the cylinder (including the base and the top) is given by
S = 2π rh + 2π r ^{2} = 2π r ( h + r )
The surface area T of the cylinder (not including the base or the top) is
T = 2π rh
Volume Of Right Circular Cylindrical Solid
Suppose that we have a cylinder as defined above (see Fig. 438). The volume V of the corresponding right circular cylindrical solid is given by
V = π r ^{2} h
Fig. 438 Surface area and volume of a right circular cylinder.
Surface Area Of General Cylinder
Suppose that we have a general cylinder whose base is any enclosed plane curve. Let A be the interior area of the base of the cylinder (thus also the interior area of the top). Let B be the perimeter of the base (thus also the perimeter of the top). Let h be the height of the cylinder, or the perpendicular distance separating the planes containing the top and the base. Let x be the angle between the plane containing the base and any line segment PQ connecting corresponding points P and Q in the top and the base, respectively. Let s be the slant height of the cylinder, or the length of line segment PQ (Fig. 439). Then the surface area S of the cylinder (including the base and the top) is
S = 2 A + Bh
The surface area T of the cylinder (not including the base or the top) is
T = Bh

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