Of Solids bounded by Planes. 15. A pyramid whose base is a triangle, is called a triangular pyramid: if the base is a quadrangle, it is called a quadrangular pyramid,—if a pentagon, it is called a pentagonal pyramid-if the base is a hexagon, it is called a hexagonal pyramid, &c. 16. In every prism, the sections formed by parallel planes, are equal polygons. Thus, if the prism, whose base is the polygon ABCDE, be cut v by the planes NP and SV, parallel to the base, the polygons NP, and SV, will each be equal to the base. E D Q B P 17. If a pyramid be cut by a plane parallel to the base, the section will be similar to the base. Thus, if the pyramid S-ABCDE, be cut by the plane abcde parallel to the base, the polygon abcde will be a figure similar to the base. E B QUEST.-15. What is a triangular pyramid? What is a quadrangular pyramid? What is a pentagonal pyramid? What a hexagonal pyramid ? 16. If a prism be cut by a plane, parallel to the base, how will the section compare with the base? 17. If a pyramid be cut by a plane parallel to ne base, will the section be similar to the base? E B 19. The convex surface of a regular pyramid, is equal to the perimeter of the base, multiplied by half the slant height. Thus, the convex surface of the pyramid S-ABCDE, is equal to SF (AB+BC+CD+DE+EA.) QUEST.-18. What is the convex surface of a right prism equal to ? 19. What is the convex surface of a regular pyramid equal to ? 20. What is the content of a solid ? Of Solids bounded by Planes. suppose AB=4 feet, and BC=3 feet. Then the num ber of square feet in the base will be equal to 3×4=12 square feet. Therefore, 12 equal cubes of one foot each, may be placed by the side of each other on the base. If the parallelopipedon be 1 foot in height, it will contain 12 cubic feet: were it 2 feet in height, it would contain two tiers of cubes, or 24 cubic feet: were it 3 feet in height, it would contain three tiers of cubes, or 36 cubic feet. Therefore, the solid content of a parallelopipedon is equal to the product of its length, breadth, and height. The 21. It may be remarked here, that there are three kinds of quantity in geometry,--viz. Lines, Surfaces, and Solids. Each of these has its own unit. unit of a line, which we have called the linear unit ( V. Art. 1), is a line of a known length, as a foot, a yard, a rod, &c. QUEST.-If one side of the base of a parallelopipedon be 4 and the other 3 feet, how many square feet will it contain? How many cubes of 1 foot each, may be placed on the base? If the parallelopipedon be 1 foot in height, how many cubes will it contain? If it be 2 feet in height, how many would it contain? If it were 3 feet in height, how many would it contain? What then, is the solid content of a parallelopipedon equal to ? 21. How many kinds of quantity are there in geometry? What are they? What is the unit of measure of a line? If a line be 30 yards long, what is the unit of measure? Of Solids bounded by Planes. The unit of surface, is a square, whose sides are the unit of length. The unit of solidity is a cube whose edges are the unit of length. For example, if the bounding lines of a surface be estimated in yards, the content will be square yards; and if the bounding lines of a solid be yards, its surface will be estimated in square yards, and its solid content in cubic yards. 22. The solid content of any prism, is equal to the area of the base, multiplied by the altitude. 23. The solid content of a pyramid, is equal to the area of the base, multiplied by one third of the altitude; or, equal to one third of the product of the base, multiplied by the altitude. QUEST.-What is the unit of measure for surfaces? What is the unit of measure for solids? If the linear unit be 1 yard, what will be the unit of surface? What the unit of solidity? 22. What is the solid content of a prism equal to ? 23. What is the solid content of a pyramid equal to ? Of the Regular Solids. SECTION XI. OF THE FIVE REGULAR SOLIDS. A Regular solid, is one whose faces are all equal polygons, and whose solid angles are equal. There are five such solids. 1. The Tetraedron, or equilateral pyramid, is a solid bounded by four equal triangles. QUEST.—What is a regular solid? How many regular solids are there 1 1. What is the regular Tetraedron or equilateral pyramid ? |