14. Volumes. Let us now examine the size of the cube. If your cube were solid and yet made of something which could be cut easily, and each edge were divided into five equal parts, the cube could be cut into layers, and each layer could be cut into a number of small cubes. 59. Can you see how many layers there would be? 60. Can you see how many small cubes there would be in each layer? 61. Can you calculate how many small cubes there would be in the whole figure? Each of these smaller cubes is called a cubic centimetre, which means a cube whose edges are one centimetre long. There are a number of cubic centimetres indicated in the figure, but you would not find great difficulty in making yourself a cubic centimetre out of paper, using the diagram at the head of this chapter. 62. How many cubes of the size you have made would it take to form a cube with edges twice as long? Perhaps you can collect the cubes from your classmates, and try the experiment by placing them together. 63. How many cubes would it take to form a cube with an edge three times as long as the first cube? 64. How many cubic centimetres are there in a cube whose edge is 2 cm. long? 65. How many, if the edge is 3 cm. long? In these questions you have been finding the volumes of cubes. The volume of a cube is the number of cubic centimetres, metres, inches, feet, etc., into which it could be divided. 66. Can you give a rule for calculating the volume of a cube when you know its dimensions? ANI I. THIS figure is called a parallelopiped (par-al-lel-o-pi'ped), a word which means "having flat, parallel surfaces." The figure has six faces like the cube, which in fact is one kind of parallelopiped; but the name is usually given to figures some at least of whose faces are not squares. If you observe the face which is turned towards us, you will see that like a square it has four edges, meeting one another perpendicularly; but unlike those of the square the edges are of two different lengths, the opposite ones being equal. Such a face is called a rectangle, which means "having right angles." Draw a square with an edge of any length, say 5 cm. (or 2 in.) and cut it out from the paper. With the aid of your folded measuring paper, rule a line straight Then cut across, perpendicular to the edges it meets. the square into two parts along the line you have just drawn. Each of these parts will be a rectangle. You will notice that the opposite edges of each rectangle are parallel ; and if you fold the rectangle over so that the opposite edges may come together, you will see that they are equal. Also you can cut each rectangle into smaller rectangles, being careful to make the dividing lines perpendicular to the edges they meet; and you can turn a rectangle into a square by cutting off the right amount. The parallelopiped is by far the commonest of all forms in architecture, occurring repeatedly in the parts of buildings. The picture of Faneuil Hall, for example, shows five distinct parallelopipeds, —three in the body of the building, one in the chimney, and one in the base of the cupola. All the faces, except two in the cupola, are rectangles; so we have here five rectangular parallelopipeds. We will now make a model of the parallelopiped. 2. The diagram will need paper 25 cm. 5 mm. X 21 cm. (or 10 X 81⁄2 in.). AB and CD are each 2 dcm. 5 cm. (or 10 in.) long and I dcm. (or 4 in.) apart; that is, AC and BD are each 1 dcm. (or 4 in.) long. AB and CD are each divided into parts beginning at A and C as follows: 5 cm. (or 2 in.), 7 cm. 5 mm. (or 3 in.), 5 cm. (or 2 in.), and 7 cm. 5 mm. (or 3 in.). EF and GH are each 2 dcm. (or 8 in.) long, extending at each end 5 cm. (or 2 in.) beyond AB and CD, which they cross at the first and second points of division from A and C. EG and FH are each 7 cm. 5 mm. (or 3 in.) long. Wherever the lines cross they are perpendicular to each other. 3. 1. How many faces has this figure? 2. How many edges? 3. How many corners? 4. If you place the figure with any face horizontal, will any other faces also be horizontal? If so, how many? |