PROPOSITION VII. THEOREM

558. Two rectangular parallelopipeds having equal bases are to each other as their altitudes.

M'

M

B

A

Hyp. AB and CD are the altitudes of parallelopipeds M and M' which have equal bases.

To prove

M: M':: AB: CD.

CASE I. When the altitudes are commensurable.

HINT. Apply common measure to altitudes AB and CD, and through the points of division pass planes | to bases. Use method of Prop. I,

CASE II.

A

When the altitudes are incommensurable.

HINT. - Divide AB into any number of equal parts, apply one of these to CD, leaving a remainder ED. Through E pass a plane. Use method of Prop. I, Bk. IV.

559. DEF. The dimensions of a rectangular parallelopiped are the three edges that meet at the same vertex.

560. SCHOLIUM. The preceding theorem may be stated: Two rectangular parallelopipeds which have two dimensions in common are to each other as the third dimension.

PROPOSITION VIII. THEOREM

561. Two rectangular parallelopipeds having equal altitudes are to each other as their bases.

Hyp. a, b, c and a, b', c' are the three dimensions of rectangular parallelopipeds M and N, respectively; a being the equal altitude.

Proof. Construct O a rectangular parallelopiped with dimensions a, b', and c.

Then

and

MOON

(Why?)

562. SCHOLIUM. This may be stated: two rectangular parallelopipeds that have one dimension in common are to each other as the products of their other two dimensions.

Ex. 1090. Two rectangular parallelopipeds have equal altitudes and bases whose dimensions are 4 and 7, and 5 and 9 respectively. Find the ratio of their volumes.

Ex. 1091. What is the ratio of the volumes of two rectangular parallelopipeds having equal bases and altitudes a and b respectively?

PROPOSITION IX. THEOREM

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

Hyp. a, b, c and a', b', c' are the three dimensions of the rectangular parallelopipeds M and N respectively.

To prove

HINT. - Construct O a rectangular parallelopiped with dimensions a, b', and c. Find the ratios and (Props. VII and VIII), and mul

M

N

tiply them together.

PROPOSITION X. THEOREM

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

M

a

Hyp. a, b, c are the dimensions of a rectangular parallelopiped M.

To prove

HINT.

volume of Maxbx c.

Construct V unit of volume, and apply Prop. IX.

=

565. COR. 1.

edge.

The volume of a cube equals the cube of its

566. COR. 2. The volume of a rectangular parallelopiped equals the product of its altitude by its base.

EXERCISES

Ex. 1092. What is the ratio of the volumes of two rectangular parallelopipeds whose dimensions are 6, 8, and 9 and 6, 7, and 9 respectively?

Ex. 1093. Find the area of the base of a rectangular parallelopiped whose altitude is 6 and one edge of the base is 4, if the solid is equivalent to a rectangular parallelopiped whose dimensions are 8, 12, and 15.

Ex. 1094. Find volume of a rectangular parallelopiped the dimensions of whose base are 12 and 20, and the area of whose entire surface is 800. Ex. 1095. Find the volume of a rectangular parallelopiped the dimensions of whose base are 12 in. and a in., and the area of whose entire surface is (120 + 34 a) sq. in.

PROPOSITION XI. THEOREM

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

Hyp. AC' is any parallelopiped whose base is ABCD and

altitude H.

To prove

volume AC" = ABCD × H.

Proof. Produce the edge AB and the edges || to it. Take EF equal to AB, and through E and F pass planes EK' and FG' EF, forming the right parallelopiped EG'.

Then

EG' AC'.

(Why?)

Produce the edge KE and the edges || to it. Take LM equal to KE, and through L and M pass planes LP' and MO' 1 LM, forming the parallelopiped LO'.

Also the planes of the upper and lower bases are parallel, and hence the three solids have a common altitude H.

By construction the planes MK' and OG' are perpendicular to plane MG, and the planes LP' and MO' are also perpendicular to plane MG', hence LO' is a rectangular parallelopiped. Now

volume LO' = LMOP × H.

Hence, from (1) volume AC' = LMOP × H,

and by substitution from (2)

volume AC' = ABCD × H.

(564)

Q.E.D.

Ex. 1096. Find the volume of a parallelopiped whose base is 50 and whose lateral edge is 20, if the inclination of the lateral edge to the base is 30°.

Ex. 1097. Find the volume of a parallelopiped whose base is 20 and whose lateral edge is 10, if the inclination of the lateral edge to the base is 45°.

Ex. 1098. In the diagram for Prop. XI, find the volume of AC', if AB = 4, BC = 5, AA' = 6, ≤ ABC = 30°, and the inclination of AA' to the base AC is 60°.