of volume.

Hence the whole solid contains 3 × 4 × 5, or

60 cubes, each equal to the unit of volume.

608. COR. 1. Since a X b is the area of the base, and c is the altitude, of the parallelopiped P; therefore the above result may be expressed in the form:

The volume of a rectangular parallelopiped is equal to the product of its base and altitude.

609. COR. 2. The volume of a cube is the third power of its edge, being the product of three equal factors; if the edge is 1, the volume is 1 x 1 X 1=1; if the edge is a, the volume is a Xax a = a'. Hence it is that in arithmetic and algebra the "cube" of a number is the name given to the "third power" of a number.

EXERCISES.

1. Find the surface, and also the volume, of a rectangular parallelopiped whose edges are 4, 7, and 9 feet.

2. Find the surface of a rectangular parallelopiped whose base is 8 by 12 feet and whose volume is 384 cubic feet. 3. Find the volume of a rectangular parallelopiped whose surface is 208 and whose base is 4 by 6.

4. Find the length of the diagonal of a rectangular parallelopiped whose edges are 3, 4, and 5.

5. Find the ratio of two rectangular parallelopipeds whose dimensions are 1, 4, 8, and 3, 4, 5, respectively.

6. Find the surface of a rectangular parallelopiped whose base is 7 by 9 feet and whose volume is 315 cubic feet.

7. Find the volume of a rectangular parallelopiped whose surface is 416 and whose base is 4 by 12.

8. A man wishes to make a cubical cistern whose contents are 186624 cubic inches: how many feet of inch boards will line it?

9. Find the side of a cube which contains as much as a rectangular parallelopiped 20 feet long, 10 feet wide, and 6 feet high,

Proposition 12. Theorem.

610. The volume of any parallelopiped is equal to the

product of its base and altitude.

The rt. parallelopiped EG', formed by the sections EE'H'H, FF'G'G to the produced edge EF is equivalent to AC'.

(590)

Now produce the edges HE, GF, G'F', H'E', and take KL = HE.

The parallelopiped KNN'K'-L formed by the sections LMM'L', KNN'K' to the produced edge KL is equivalent to EG', and ... to AC'.

(590)

... given parallelopiped AC' and the last, KM', are equivalent.

But, since the rt. sections LM', KN', are rectangles,

... KM' is a rectangular parallelopiped.

Also, since

(582)

area ABCD = area EFGH = area KLMN, (363) (360), and the three solids have the same altitude B'O,

... the volume KM' = KLMN × B'O.

... the volume AC' ABCD × B'O.

(523)

(608)

Q.E.D..

Proposition 13. Theorem.

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

Hyp. Let H denote the altitude of

the triangular prism ABC-B'.

To prove vol. ABC-B' ABC x H.

=

D'

B

Proof. Complete the parallelopiped D ABCD-D', having its edges and || to

=

A

AB, BC, BB'.

B

Then,

(600)

612. COR. 1. The volume of any prism is equal to the product of its base and altitude.

For, any prism may be divided into triangular prisms by passing planes through a lateral edge AA' and the corresponding diagonals of the base. Then the volume of the given prism is the sum of the volumes of the triangular prisms, that is, the sum of their bases, or the base of the given prism, multiplied by the common altitude.

A

ய

B

D'

613. COR. 2. Two prisms are to each other as the products of their bases and altitudes; two prisms having equivalent bases are to each other as their altitudes; two prisms of the same altitude are to each other as their bases; two prisms having equivalent bases and equal altitudes are equivalent.

PYRAMIDS.

DEFINITIONS.

614. A pyramid is a polyedron bounded by a polygon, and by triangles meeting at a common point, the sides of the polygon being the bases of the triangles.

The polygon ABCDE is called the base of the pyramid; the point S, in which the triangles meet, is called the vertex; the triangular faces are called the lateral faces, and taken together they form the lateral, or convex, surface; the intersections SA, SB, etc., of the lateral faces are called the lateral edges; and the area of the lateral surface is called the lateral area.

S

615. The altitude of a pyramid is the perpendicular distance from the vertex to the plane of the base.

616. A pyramid is called triangular, quadrangular, pentagonal, etc., according as its base is a triangle, quadrilateral, pentagon, etc.

617. A triangular pyramid has but four faces, and is called a tetraedron; any one of its faces can be taken for its base.

NOTE. The six edges of a triangular pyramid may be divided into three pairs, such that the two edges of a pair do not meet each other. Since each edge meets two other edges at one vertex, and yet another two edges at the adjoining vertex, there is but one edge left to pair with it. The pair is called a pair of opposite edges.

618. A regular pyramid is one whose base is a regular polygon, the centre of which coincides with the foot of the

perpendicular let fall on it from the vertex. This perpendicular is called the axis of the pyramid.

619. The lateral edges of a regular pyramid are equal (496); therefore the lateral faces are equal isosceles triangles.

620. The slant height of a regular pyramid is the perpendicular distance from the vertex to the base of any one of its lateral faces.

621. A frustum of a pyramid is the portion of a pyramid included between its base and a plane parallel to the base, and cutting all the lateral edges. If the cutting plane is not parallel to the base, the portion included between the base and plane is called a truncated pyramid.

622. The altitude of a frustum is the perpendicular distance between its bases.

In a frustum of a regular pyramid the lateral faces are equal trapezoids.

The slant height of the frustum of a regular pyramid is the perpendicular distance between the parallel sides of one of these trapezoids..

EXERCISES.

1. Show that the slant height of a regular pyramid is the straight line drawn from the vertex of the pyramid to the middle point of any side of the base.

2. Show that the lateral faces of a frustum of a regular pyramid are equal trapezoids.

3. Find the volume of a right triangular prism if the height is 24 inches and the sides of the base are 24, 20, and 20 inches.

4. Find the volume of a triangular prism if the height is 18 inches and the sides of the base are 6, 8, and 10 inches.