760. Similar Polyedrons. Two polyedrons are similar if they have the same number of faces similar each to each and similarly placed, and have their corresponding polyedral angles congruent. It is evident that the corresponding diedral angles are also equal. 761. Theroem. The corresponding edges of similar polyedrons are proportional. E H' Fi G H G Corresponding edges belonging to corresponding faces are proportional by § 444. Use the fact that each edge belongs to two faces and prove the theorem. 762. Theorem. The surfaces of two similar polyedrons are to each other as the squares of any two corresponding edges. Given two similar polyedrons P and P', with corresponding faces A and A', B and B', C and C',.... Also given a and a', b and b', c and c',..., corresponding sides of these faces respectively. 763. Theorem. The volumes of two similar tetraedrons are to each other as the cubes of any two corresponding edges. Use §§ 746, 761. 764. Theorem. The volumes of two similar polyedrons are to each other as the cubes of any two corresponding edges. This theorem will not be proved. It is accepted as true since it has useful applications. EXERCISES 1. What is the ratio of the volumes of two cubes that are 5 in. and 4 in. on an edge? What is the ratio of their areas? 2. The edges of a parallelepiped are 10, 12, and 14. Find the edges of a similar parallelepiped having as great an area. their volumes. Find the ratio of 3. Two pyramids are cut from the same pyramidal surface. The lateral edges of one are 9 in., 12 in., and 14 in. and of the other 18 in., 21 in., and 20 in. Find the ratio of their volumes. 4. Find the ratio of the volume of a regular octaedron 6 in. on an edge to the volume of a regular octaedron having half as great an area. 5. The center of gravity of a tetraedron is its altitude above the base. Find the center of gravity of a regular tetraedron 8 in. on an edge. 6. The edge of a regular tetraedron is e. Find the edge of a regular tetraedron that has a volume n times as great as the volume of the given tetraedron. 7. Find the edge of a regular tetraedron such that its volume multiplied by √2 is 288. 8. Find the volume of a regular tetraedron that has an altitude of 17 in. 9. A section of a tetraedron by a plane parallel to two opposite edges is a parallelogram. 10. The midpoints of the edges of a regular tetraedron are the vertices of a regular octaedron. 11. Two similar polyedrons have volumes of 121.5 cu. in. and 4.5 cu. in., respectively. An edge of the smaller is 11⁄2 in. Find the corresponding edge of the larger. Ans. 4.5 in. 12. The area of the entire surface of a polyedron is 108 sq. in., and its volume is 432 cu. in. If the area of the entire surface of a similar polyedron is 75 sq. in., find its volume. Ans. 250 cu. in. GENERAL EXERCISES 1. If from any point within a regular tetraedron perpendiculars are drawn to its faces, their sum equals an altitude of the tetraedron. (Compare this with Ex. 2, p. 96.) 2. A wedge whose altitude is 10 in. and edge 4 in., has a base that is a square having a perimeter of 36 in. Find the volume of the wedge. 3. A concrete pier for a railway bridge has dimensions as shown in the figure, the bases being rectangles with semicircles. Find the number of cubic yards of concrete in the pier. Ans. 37.8 cu. yd. 4. A railroad cut has the dimensions given in the figure, which shows the vertical section and three cross sections, one at each end and one in the middle. Find the number of cubic yards of earth removed in digging the cut. Ans. SUGGESTION. 7972 cu. yd. 22 12 11 5 150 yd. -150 yd. 20 + The part on each side of the center is a prismatoid, and can be computed separately. 5. A tank of reinforced concrete is 160 ft. long, 100 ft. wide, and 10 ft. 6 in. deep, outside dimensions. The side walls are 8 in. thick at the top and 18 in. thick at the bottom, with the slope on the inside. The bottom is 6 in. thick. Find the number of cubic yards of cement used in building the tank, and the capacity of the tank in gallons. 6. The vertices of one regular tetraedron are the centers of the faces of another regular tetraedron. Find the ratio of the volumes of the two tetraedrons. 7. The three planes passing through the lateral edges of a triangular pyramid, and bisecting the base edgcs, meet in a common straight line. 8. The six planes passing through the edges of a tetraedron, and bisecting the opposite edges, meet in a common point. 9. The point of intersection of the planes in the preceding exercise is the center of gravity of the tetraedron. Prove that the center of gravity of a tetraedron divides the line from a vertex to the center of gravity of the opposite face in parts that are in the ratio of 3: 1. The center of gravity of a face is at the intersection of the medians of that face, 10. The corresponding edges of three similar tetraedrons are 3 in., 4 in., and 5 in., respectively. Find the corresponding edge of a similar tetraedron equal in volume to their sum. 11. By means of the prismatoid formula show that the volume of a truncated quadrangular prism whose opposite faces are parallel, is equal to the area of a right section times one-fourth the sum of the lateral edges. SUGGESTION. Consider two of the parallel faces, as B1 and B2, as bases. Then M is the midsection between these. Let A be the area of a right section having a side in B1 equal to n, and altitude h, the distance between B1 and B2. Then B1 = 1⁄2 (a+d) n, B2 = 1⁄2) (b+c) n, and M = 1⁄2 { 1⁄2 ·. V = {h [1⁄2 (a+d) n+1⁄2 (b+c) n+2 { 1⁄2 (a+b) + a B1d M (a+b) + 1⁄2 (c+d)} n (c+d) } n] = {hn (a+b+c+d) = 1A (a+b+c+d). 12. The bottom of a bin of wheat is a rectangle 5 ft. by 12 ft. The top of the wheat is in a plane such that the depths at the corners are 6 ft., 5 ft., 3 ft., and 4 ft. respectively. Find the number of bushels in the bin if 1 bu. = 1 cu. ft. CHAPTER X THE SPHERE AND POLYEDRAL ANGLES GENERAL PROPERTIES 765. Sphere. A sphere is a closed surface all points of which are equally distant from a point within, called the center of the sphere. A sphere is usually designated by a single letter at its center. The radius of a sphere is the straight line connecting the center to a point in the surface. A straight line 'connecting two points in the surface and also passing through the center is a diameter. A sphere separates space into two parts so that any point that does not lie in the surface lies within the sphere or outside it. The distance from the center to a point within a sphere is less than the length of the radius; while a point outside a sphere is at a greater distance from the center than the length of the radius. 766. Great Circle of a Sphere. It is evident from the definitions of a sphere and of a circle that a plane passing through the center of sphere intersects the sphere in a circle whose radius is equal to the radius of the sphere. Such a circle is called a great circle of the sphere. |