Proposition 255 Theorem

Two rectangular parallelepipeds are to each other as the products of their three dimensions.

Let the parallelepipeds P and P' have the dimensions a, b, and c and a', b', and c' respectively. Compare P and P' with a third parallelepiped whose dimensions are a, b, and c'.

Proposition 256 Theorem

The volume of a rectangular parallelepiped is equal to the product of its three dimensions.

HINT. Find the ratio of the parallelepiped to the cubic unit taken as the unit of volume.

COR. I. The volume of a rectangular parallelepiped is equal to the product of its base and altitude.

COR. II. The volume of a cube is equal to the cube of its edge. For this reason the third power of a quantity is called the cube of the quantity. The volume of a cube having the line AB as an edge is written AB.

DISCUSSION. When the three dimensions are of such lengths that the linear unit is contained an exact number of times in each, this proposition is rendered evident by dividing the rectangular parallelepiped

into cubes, each equal to the unit of volume. For example, if the three dimensions are 4 linear units, 3 linear units, and 2 linear units, the parallelepiped may

be divided into 24 cubes, each equal to the unit of volume; that is, the volume is equal to 4 X 3 X 2 units of volume.

The volume of any parallelepiped is equal to the product of its base and altitude,

Consult Prop. 252 and Prop. 256.

The volume of a triangular prism is equal to the product of its base and altitude.

Hypothesis. Let V denote the volume, B the base, and h the altitude of the triangular prism ACD-E.

Proof. Construct the parallelepiped ACDH-F, having AC, CD, and CF as edges.

The volume of any prism is equal to the product of its base and altitude.

HINT. Let the prism be divided into triangular prisms by the planes determined by a lateral edge and the diagonals of the base drawn from the foot of this edge. Consult Prop. 258 and Ax. 4.

V = BX h.

COR. I. Prisms having equivalent bases and equal altitudes are equivalent.

COR. II. Prisms having equivalent bases which lie in parallel planes are equivalent.

COR. III. Any two prisms are to each other as the products of their bases and altitudes.

COR. IV. Two prisms having equivalent bases are to each other as their altitudes.

COR. V. Two prisms having equal altitudes are to each other as their bases.

Ex. 1084. The volume of an oblique prism is equal to the area of a right section and the length of a lateral edge.

Ex. 1085. The volume of a triangular prism is equal to one half the product of the area of a lateral face and the distance of that face from the opposite lateral edge.

Ex. 1086. The volume of a regular prism is equal to one half the product of the lateral area and the apothem of the base.

Ex. 1087. Show how to divide an oblique prism into two equivalent parts by passing a plane through it parallel to the bases.

Ex. 1088. Show how to divide a regular prism into two equivalent parts by passing a plane through it parallel to the lateral edges.

PYRAMIDS

A pyramidal surface is a surface generated by a straight line which continually moves along a fixed broken line and passes through a fixed point not coplanar with the broken line. The moving line is called the generatrix; the broken line is called the directrix; and the fixed point is called the vertex.

It follows from the definition that a pyramidal surface is composed of planes, the intersections of which are concurrent. The two parts on opposite sides of the vertex are called the upper and lower nappes.

A pyramid is a polyhedron bounded by a closed pyramidal surface and a plane cutting the generatrix in every position. The section of the pyramidal sur

face formed by the plane is called the base of the pyramid, the faces formed on the pyramidal surface are called the lateral faces, the intersections of the lateral faces are called the lateral edges, and the sum of the areas of the lateral faces is called the lateral area.

NOTE. Only pyramids having convex polygons for bases are considered in this book.

A pyramid is called triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrilaterals, pentagons, etc.

NOTE. A triangular pyramid is a tetrahedron, and any one of its faces may be taken as its base.

The perpendicular distance from the vertex to the base is called the altitude of the pyramid.

A regular pyramid is a pyramid having for its base a regular polygon, the centre of which is the foot of the altitude.

A truncated pyramid is the part of a pyramid included between the base and a section cutting all the edges. The base of the pyramid and the section thus made are respectively the lower base and the upper base of the truncated pyramid.

A frustum of a pyramid is a truncated

pyramid whose bases lie in parallel planes. The perpendicular distance between the bases of a frustum is called the altitude.

The following principles follow at once from the preceding definitions:

(i) The lateral edges of a regular pyramid are equal.

(ii) The lateral edges of a frustum of a regular pyramid are equal.

(iii) The lateral faces of a pyramid are triangles.

(iv) The lateral faces of a regular pyramid are congruent isosceles triangles.

(v) The altitudes of the lateral faces of a regular pyramid are equal.

(vi) The lateral faces of a frustum of a pyramid are trapezoids.

(vii) The lateral faces of a frustum of a regular pyramid are congruent isosceles trapezoids.

(viii) The altitudes of the lateral faces of a frustum of a regular pyramid are equal.

DEFINITIONS. The altitude of any lateral face of a regular pyramid is called the slant height of the pyramid.

The altitude of any lateral face of a frustum of a regular pyramid is called the slant height of the frustum.