Surface Area and Volume for Physics Help

Introduction

Let’s go from two dimensions to three. Here are some formulas for surface areas and volumes of common geometric solids. The three-space 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. 4-35). The volume V of the pyramid is given by

V = Ah /3

Fig. 4-35 Volume of a pyramid.

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. 4-36). 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. 4-36 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. 4-37). 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

Fig. 4-37 Volume of a general conical solid.

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. 4-38). 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

Fig. 4-38 Surface area and volume of a right circular cylinder.

Volume Of Right Circular Cylindrical Solid

Suppose that we have a cylinder as defined above (see Fig. 4-38). The volume V of the corresponding right circular cylindrical solid is given by

V = π r ² h

Fig. 4-38 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. 4-39). 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

Fig. 4-39 Surface area and volume of a general cylinder and an enclosed solid.