(3) A parallelepiped has three sets of four equal edges.

(4) Any two opposite faces of a parallelepiped may be taken as bases.

(5) All the faces of a rectangular parallelepiped are rectangles. (6) All the faces of a cube are squares.

671. Volume of Rectangular Parallelepiped. By a consideration analogous to that of § 344 the reasonableness of the following statement would be evident; here it is accepted without proof.

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

If V, a, b, and h are the numerical measures of the volume, length, breadth, and height, respectively, of any rectangular parallelepiped, then the above is stated in the formula

V = abh.

This means that, if the three dimensions are

each measured in terms of some unit of length, h then the volume is measured in terms of the corresponding unit of volume, which is a cube having an edge one linear unit in length.

α

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

673. Theorem. The volume of a cube is equal to the cube of its edge.

674. Theorem. The volumes of two rectangular parallelepipeds are to each other as the products of their three dimensions. V: V'abh: a'b'h'.

675. Theorem. The volumes of two rectangular parallelepipeds having equivalent bases are to each other as their altitudes.

676. Theorem. The volume of two rectangular parallelepipeds having equal altitudes are to each other as their bases. 677. Theorem. Two rectangular parallelepipeds having equivalent bases and equal altitudes are equal in volume.

1. Find the volumes (1) 13 ft. by 27 ft. by 45 ft.

EXERCISES

of the following rectangular parallelepipeds: (2) 2 ft. 3 in. by 1 ft. 7 in. by 3 ft. 1 in. (3) 30 ft. 6 in. by 41 ft. 6 in. by 12 ft.

2. A common brick is 8 in. by 4 in. by 2 in. Find the number of common brick in a pile 3 ft. by 6 ft. by 12 ft.

3. How many shoe boxes each 3 in. by 4 in. by 9 in. can be put in a packing box 3 ft. by 3 ft. 4 in. by 3 ft. 9 in.?

4. Find the number of cubic yards of earth to be excavated in digging a cellar 40 ft. by 26 ft. by 7 ft.

5. If 180 sq. ft. of zinc are required to line the bottom and sides of a cubical vessel, how many cubic feet of water will it hold?

6. A box car that is 363 ft. long and 8 ft. wide, inside measurements, can be filled with wheat to a height of 41 ft. Find how many bushels of wheat it will hold if 1⁄2 cu. ft. are a bushel.

7. Find the cost at 40 cents a pound for sheet copper to line the bottom and sides of a cubical vessel 7 ft. on an edge, if the sheet copper weighs 12 oz. per square foot. Find volume. Ans. $73.50; 81.45 bbl.

8. Find the cost of laying a stone wall 45 ft. long, 6 ft. high, and 2 ft. thick at $2.75 a perch. Use 22 cu. ft. for one perch.

9. The edge of a cube is 10 in. Find the edge of a cube which shall have a volume twice as great. Eight times as great.

10. Show that the edge and diagonal of a cube can be used as the two sides of a right triangle whose acute angles are 30° and 60°.

11. Are the diagonals of a rectangular parallelepiped perpendicular to each other? Are those of a cube? Prove.

12. The diagonals of a rectangular parallelepiped are equal.

13. The diagonals of a rectangular parallelepiped bisect each other. 14. Find the sum of all the face angles of a parallelepiped.

15. Find the edge of a cube if its volume is increased 200 cu. in. when each edge is increased 2 in.

16. The edges of a rectangular parallelepiped are a, b, and c. Find the length of a diagonal, and the entire area of the parallelepiped.

17. Test the following rule if 1 cu. in. of iron weighs 0.28 pounds: The weight of iron bars in pounds per foot of length equals the width in inches times the thickness in inches times.

18. The sum of the squares of the four diagonals of any parallelepiped is equal to the sum of the squares of the twelve edges. Apply Ex. 11, p. 181.

678. Theorem. The volume of any parallelepiped is equal to the product of its base and altitude. V=Bh.

Given X an oblique parallelepiped, with area of base AC-B, and height h.

To prove V=Bh, where V denotes volume.

Proof. Extend DC and the other edges of X that are parallel to it.

On DC extended, take EF=DC.

Pass planes EG and FH through E and F and perpendicular to DC, forming a right parallelepiped Y, in which EG is a right section.

Then parallelepiped X = parallelepiped Y.

$660

Extend IE and the other edges of Y that are parallel to it. On IE extended, take JK=IE.

Pass planes JL and KM through J and K and perpendicular

to IE, forming a rectangular parallelepiped Z.

Then parallelepiped Y = parallelepiped Z.

Therefore parallelepiped X = parallelepiped Z.

But volume of Z=base JN×NM.

§ 660

§ 104

§ 672

Why?

Why?

Why?

... V=Bh.

679. Theorem. Parallelepipeds having equivalent bases and

equal altitudes are equivalent.

680. Theorem. Two parallelepipeds are to each other as the products of their bases and altitudes. ·

681. Theorem. Two parallelepipeds having equal altitudes are to each other as their bases.

682. Theorem. Two parallelepipeds having equivalent bases are to each other as their altitudes.

B=

EXERCISES

1. Compare theorems of §§ 678-682 with those of §§ 355, 358-361. 2. Use the notation of § 678, and show that in any parallelepiped

3. Find the length of the diagonal of a rectangular parallelepiped whose dimensions are 30 ft., 40 ft., and 12 ft.

4. A parallelepiped has a base in the form of a rhombus whose edges and one diagonal are each 10 in. Find the volume if the altitude is 8 in. 5. A parallelepiped has a rectangular base 8 in. by 15 in., and square ends. Find its volume if one of the sides is a parallelogram having an angle of 60°.

6. Find the diagonal of a cube whose volume is 512 cu. in.

7. Find the dimensions of a rectangular parallelepiped having a volume of 12,960 cu. in., if the dimensions are in the ratio of 3:4:5. Ans. 18 in., 24 in., 30 in.

8. Find the edge of a cube whose surface and volume have the same numerical value.

9. A parallelepiped of altitude 8 in. has the same volume as a cube that is 12 in. on an edge. Find the area of the base and the dimensions of the base if it is a parallelogram having an angle of 60° and one side equal to 18 in.

10. Find the length of the bar that can be made from 1 cu. ft. of steel, if the bar has a rectangular cross section in. by 11⁄2 in.

11. Find the volume to the nearest .001 cu. ft. of a rectangular tank 17 ft. 2 in. by 19 ft. 3 in. by 3 ft. 7 in. Ans. 1184.142 cu. ft.

12. Show geometrically that (a+b)3=a3+3a2b+3ab2+b3.

13. The total area of a right prism whose base is a rectangle is 118 sq. ft., the volume is 70 cu. ft., and the altitude is 7 ft. Find dimensions of the base.

VOLUMES OF PRISMS AND CYLINDERS

683. Theorem. The plane passed through the two diagonally opposite edges of a parallelepiped divides the parallelepiped into two equivalent triangular prisms.

Given the parallelepiped AC', divided into two triangular prisms A'-ABD and C'-BCD by the diagonal plane passing through the edges BB' and DD'.

To prove prism A'-ABD=prism C'-BCD.

Proof. Let EFGH be a right section of the parallelepiped.

cutting the diagonal plane in FH.

Face AB' face DC', and face AD' face BC'.

Then EF HG, and EH || FG.

And EFGH is a parallelogram.

Hence

FH is a diagonal of EFGH.

AEFH AFGH.

Why?

§ 562

Why?

Why?

§ 154

§ 660

Prism A'-ABD equals a right prism with base EFH and altitude AA'.

Why?

Prism C'-BCD equals a right prism with base FGH and altitude AA'. But these two right prisms are congruent and so equivalent. .. prism A'-ABC=C'-BCD.

EXERCISES

§ 104

1. Find the edge of a cube that is increased in volume 127 cu. in. when its edge is increased 1 in.

2. A rectangular solid with a square base has a volume of 80 cu. ft., and a surface of 112 sq. ft. Find its dimensions. Ans. 4 ft. by 4 ft. by 5 ft.