Compare the rectangular parallelopipeds P and R, whose edges are x, y, z and x', y', z'. Sug. Construct a third parallelopiped, Q, with edges x', z, y. Write an equation comparing P and Q. Call it (1). In the same manner compare Q and R. Call this equation (2). Multiply (1) by (2) and simplify. Explain the meaning of this third equation. Generalize. Call this Prop. X. 488. Cor. Suppose that P were a cube with edges 1, 1, 1 Write the equation expressing the relation of R to P. If we call P a unit of volume, what does your equation show as to the number of units of volume in R? What expresses the volume of a rectangular parallelopiped? EXERCISE. 459. When the edges of the rectangular parallelopiped are multiples of the linear unit, construct a figure showing how to compute the volume of the solid. Given any oblique parallelopiped, A G; i. e., all angles oblique. Make E' F' = E F. Pass planes E' D', F' C′ 1 to E' F'. Sug. 1. By considering E' D' and F' C'bases, what kind of a solid have we? [$478.] How does the edge E' F' compare with E F? Compare E C and E' C'. [$ 478. ] Sug. 2. Now consider D' Produce D' A', making A' M C' G' H' as the base of A'G'. and M N L K. What kind of solid is A' L? Compare A' L with A' G'. [Prop. V., §478.] How does it compare with A G? Can you generalize the result? Call this Prop. XI. 490. Cor. I. How do find the volume of any parallelopiped? 491. Cor. II. Construct a figure and show how to find the volume of any triangular prism. 492. PROPOSITION XII. Given a prism whose base is a polygon of n sides. Into how many triangular prisms may it be divided by passing planes through the lateral edges? Construct a figure and show how to find the volume of any prism? Write the general truth, and call it Prop. XII. 493. Cor. Can you show that any two prisms are to each other as the products of their bases and altitudes? Compare two prisms having equivalent bases. Compare two prisms having altitudes. In figure Prop. XI. may we compare A G and A' G' as prisms? A G and A' L? What statement can you make concerning these prisms? of any two prisms having equivalent bases and equal altitudes? EXERCISES. 460. The dimensions of the base of a rectangular parallelopiped are 3 and 4 centimeters and the entire surface is 52 square centimeters. Find the volume. 461. The volume of a rectangular parallelopiped is 60 cubic centimeters, the entire surface 94 square centimeters, and the altitude is 3 centimeters. Find the dimensions of the base. 462. Find the lateral surface of a regular triangular prism, each side of whose base is 5 centimeters, and whose altitude is 10 centimeters. Suppose the base were 6, 5, 5 centimeters, what would be the entire surface? 463. Find convex surface and volume of a regular hexagonal prism each side of whose base is 1 dm. and whose altitude is 10 dm. 464. Two triangular prisms P and Q, have the same altitude; P has for its base a right isosceles triangle; Q has for its bise an equilateral triangle of side equal to the hypotenuse of the base of P. What is the ratio of the volumes of P and Q? 465. Find the ratio of the convex surfaces of P and Q, in Ex. 464. 466. A rectangular parallelopiped is 4, 6, and 9 centiWhat is the edge of an equivalent cube? meters. PYRAMIDS. 494. A pyramid is a polyedron bounded by a polygon, and a series of triangles which meet in a common point and whose bases are the sides of the polygon. Thus the polygon A B C D E is called the base of the pyra mid. The common vertex of the triangular faces is called the vertex of the pyramid. The edges passing through the vertex are called the lateral edges. The perpendicular from the vertex to the base is called the altitude. 495. A pyramid is triangular, quadrangular, pentagonal, etc., according as its base is a triangle, a quadrilateral, a pentagon, etc. 496. A regular pyramid has for its base a regular polygon, and its vertex lies in the perpendicular erected at the center of the polygon. 497. PROPOSITION XIII. F B Can you show how the lateral edges of a regular pyramid are related? |