When the unit of length is a foot, the unit of volume is the volume of a cube each of whose edges is a foot in length; or, as we say, the unit of volume is a cubic foot.

When the unit of length is an inch, or a yard, the unit of volume is a cubic inch, or a cubic yard.

The measure of a volume is the number which expresses how many times it will contain the unit of volume.

If in a rectangular parallelepiped, the

three edges which meet in any vertex are commensurable magnitudes, and these are divided into unit lengths, the whole volume can be divided into unit volumes by passing planes through the points of division. Thus the parallelepiped in the diagram contains thirty-six units of volume.

For brevity we frequently say "the volume of a parallelepiped" is a certain number, when it would be more exact to say "the measure of the volume."

In the preceding proposition, if the parallelepiped whose volume is should have each edge of unit length, so that

That is, a xbx c is the measure of the volume P.

503. THEOREM.

The measure of the volume of any rectangu

lar parallelepiped is the product of its three dimensions.

504. COROLLARY.

The volume of any rectangular parallele

piped is the product of its altitude and the area of its base.

PROPOSITION X

505. The volume of any parallelepiped is equal to the product of its altitude and the area of its base.

Let AG be any oblique parallelepiped whose base is ABCD and altitude KE.

It is required to prove that the volume of AG is equal to the product of KE and the area of ABCD.

Proof. Produce the edges AB, DC, HG, EF, and cut them perpendicularly at A', D', H', E', and B,' C,' G', F", by two parallel planes whose distance apart A'B' is equal to AB. Then A'G' is a right parallelepiped, A'H' being the base, equal in volume to AG. (Prop. IV.)

Again, produce the edges D'A', C'B', G'F", H'E', of the parallelepiped A'G', and cut them perpendicularly at D", C'', G", H', and A", B", F", E", by two parallel planes whose distance apart B"C" is equal to B'C'.

Then A"G" is a rectangular parallelepiped [why ?] equal in volume to A'G' (Prop. IV), and hence equal to AG.

Now the volume of A"G" is equal to the product of the altitude A"E" and the area of the base A"C". (Art. 504.)

But since all three parallelepipeds lie between the same two parallel planes, the altitude KE equals the altitude A"E".

Also the parallelograms AC, A'C', and A"C" are equal in

area.

Therefore, volume of AG= volume of A"G"

= area of A'C" × A"E"

= area of AC × KE.

(Art. 294)

PROPOSITION XI

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

E

B

Let ABC-DEF be any triangular prism whose altitude is h.

It is required to prove that the volume of this prism is equal to h times the area of the base ABC.

Proof. Complete the parallelogram ABCK. Through K draw a line parallel to a lateral edge AD, meeting the plane DEF at L, and complete the parallelepiped BL having the same altitude as the prism.

BL.

The prism ABC-DEF is one-half of the parallelepiped (Prop. VI.) Now the volume of BL is the product of its altitude h and the area of the base ABCK. (Prop. X.) Therefore the volume of the prism ABC-DEF is equal to h times the area of ABC, which is half of the base ABCK.

507. COROLLARY I. The volume of any prism is equal to the product of its altitude and the area of its base.

Any prism can be divided into triangular prisms by passing planes through one lateral edge and all the nonadjacent lateral edges.

508. COROLLARY II. The volumes of any two prisms are equal if they have equal altitudes and bases of equal areas.

EXERCISES

1. The square on the diagonal of a rectangular parallelepiped is equal to the sum of the squares on the three edges meeting in one extremity of the diagonal.

2. The square on the diagonal of a cube is three times the square on one of its edges.

3. Show that all the diagonals of a rectangular parallelepiped_are equal.

4. The sum of the squares on the four diagonals of a parallelepiped is equal to the sum of the squares on the twelve edges.

5. Every section of a prism made by a plane parallel to an edge is a parallelogram.

6. Find the volume and the length of the diagonal of a cube whose edge is 3 feet.

7. Show that all the diagonals of any parallelepiped pass through one point, and are bisected at that point.

8. The edge of a cube is 4 feet. Find the edge of a cube having twice the volume.

9. The dimensions of a rectangular parallelepiped are 3, 5, and 7 decimeters. Find its volume and the length of its diagonals.

10. A box, covered top and bottom, which is made of boards 2 inches thick, has outside dimensions of 18, 24, and 30 inches. Find its entire contents.

11. Find the volume of a right prism whose base is a regular hexagon of 6 inches side, and whose altitude is 15 inches.

12. The diagonal of a cube is 27 inches, find its volume.

SECTION II

PYRAMIDS

509. A pyramid is a polyhedron one of whose faces is a polygon, and the remaining faces are triangles having a common vertex.

The polygonal face is called the base of the pyramid, the triangular faces are the lateral faces, and the common vertex of the lateral faces is the vertex of the pyramid.

The sum of the areas of the lateral faces is called the lateral area of the pyramid.

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

A triangular pyramid is one whose base is a triangle; a quadrangular pyramid, one whose base is a quadrilateral, etc.

A triangular pyramid is a tetrahedron since it has four faces. All the faces of a tetrahedron are triangles. Any face may be taken as the base, and the opposite vertex as the vertex.

How many vertices has a tetrahedron ? how many edges? how many pairs of non-intersecting edges? These are called the pairs of opposite edges.

510. A regular pyramid is one whose base is a regular polygon and whose vertex lies on the perpendicular to the base drawn from its centre.

The slant height of a regular pyramid is the altitude of any one of its lateral faces; i.e. it is the perpendicular distance of the vertex from any side of the base.

511. A truncated pyramid is the figure formed by the base and any section of a pyramid, and the portions of the lateral faces intercepted between them.

If the upper section is parallel to the base, the figure is called a frustum of a pyramid.