A rectangular block whose length, breadth, and height are all equal is called a cube. Its surface consists of six equal squares. We will now see how models of these solids may be constructed, beginning with the cube, as being the simpler figure. Suppose the surface of the cube to be cut along the upright edges, and also along the edge HG; and suppose the faces to be unfolded and flattened out on the plane of the base. The surface would then be represented by a figure consisting of six squares arranged as below. This figure is called the net of the cube: it is here drawn on half the scale of the cube shewn in outline above. To make a model of a cube, draw its net on cardboard. Cut out the net along the outside lines, and cut partly through along the dotted lines. Fold the faces over till the edges come together; then fix the edges in position by strips of gummed paper. Ex. 3. Make a model of a cube each of whose edges is 6'0 cm. Ex. 4. Make a model of a rectangular block, whose length is 4", breadth 3", height 2". First draw the net which will consist of six rectangles arranged as below, and having the dimensions marked in the diagram. Now cut the net out, fold the faces along the dotted lines, and secure the edges with gummed paper, as already explained. (Prisms.) Let us now consider a solid whose side-faces (as in a rectangular block) are rectangles, but whose ends (i.e. base and top), though equal and parallel, are not necessarily rectangles. Such a solid is called a prism. The ends of a prism may be any congruent figures: these may be triangles, quadrilaterals, or polygons of any number of sides. The diagram represents two prisms, one on a triangular base, the other on a pentagonal base. Ex. 5. Draw the net of a triangular prism, whose ends are equilateral triangles on sides of 5 cm., and whose side-edges measure 7 cm. (Pyramids.) S B The solid represented in this diagram is called a pyramid. The base of a pyramid (as of a prism) may have any number of sides, but the side-faces must be triangles whose vertices are at the same point. The particular pyramid shewn in the Figure stands on a square base ABCD, and its side-edges SA, SB, SC, SD are all equal. In this case the side faces are equal isosceles triangles; and the pyramid is said to be right, for if the base is placed on a level table, then the vertex lies in an upright line through the mid-point of the base. Ex. 6. Make a model of a right pyramid standing on a square base. Each edge of the base is to measure 3", and each side-edge of the pyramid is to be 4". To make the necessary net, draw a square on a side of 3". This will form the base of the pyramid. Then on the sides of this square draw isosceles triangles making the equal sides in each triangle 4′′ long. Explain why the process of folding about the dotted lines brings the four vertices together. 8 3 4" Another important form of pyramid has as base an equilateral triangle, and all the side edges are equal to the edges of the base. FIG. 1. FIG. 2. FIG. 3. How many faces will such a pyramid have? How many edges? What sort of triangles will the side-faces be? Fig. 3 shews the net on a reduced scale. A pyramid of this kind is called a regular tetrahedron (from Greek words meaning four-faced). Ex. 7. Construct a model of a regular tetrahedron, each edge of which is 3" long. Ex. 8. What is the smallest number of plane faces that will enclose a space? What is the smallest number of curved surfaces that will enclose a space? The solid figure here represented is called a cylinder. On examining the model of which the last diagram is a drawing, you will notice that the two ends are plane, circular, equal, and parallel. The side surface is curved, but not curved in every direction; for it is evidently possible in one direction to rule straight lines on the surface: in what direction? Let us take a rectangle ABCD (see Fig. 2), and suppose it to rotate about one side AB as a fixed axis. What will BC and AD trace out, as they revolve about AB? Observe that CD will move so as always to be parallel to the axis AB, and to pass round the curve traced out by D. As CD moves, it will generate (that is to say, trace out) a surface. What sort of surface? We now see why in one direction, namely parallel to the axis AB, it is possible to rule straight lines on the curved surface of a cylinder. It is easy to find a plane surface to represent the curved surface of a cylinder. Cut a rectangular strip of paper, making the width PQ equal to the height of the cylinder. Wrap the paper round the cylinder, and carefully mark off the length PS that will make the paper go exactly once round. Cut off all that overlaps; and then unwrap the covering strip. You have now a rectangle representing the curved surface of the cylinder, and having the same area. |