Volume = area of base x alt. + 3 x3 = 3 – 81V3. Ex. Find the lateral edge, lateral area, and volume of a frustum of a regular hexagonal pyramid, the sides of whose bases are 14 and 6, respectively, and whose altitude is 8. Let A and O be the centers of the upper and lower bases, respectively, of the frustum of a regular hexagonal pyramid. Draw AC and OH perpendicular to BD and EG, respectively. Draw BF perpendicular to EG. HM = 0H – AC = 7V3 – 3V3 = 4V3. BE = VbF* + FE* = V(477)*+(EG – BD)*=8V2. = (294 V3 + 54 V3 + 126 V3) { = 1264 V3. Find lateral edge, area, and volume Ex. 1137. Of a regular triangular pyramid, each side of whose base is 10 and whose altitude is 18. Ex. 1138. Of a regular quadrangular pyramid, each side of whose base is 6 and whose altitude is 12. Ex. 1139. Of a regular hexagonal pyramid, each side of whose base is 6 and whose altitude is 16. Ex. 1140. Of a frustum of a regular triangular pyramid, the sides of whose bases are 12 and 4, and whose altitude is 15. Ex. 1141. Of a frustum of a regular quadrangular pyramid, the sides of whose bases are 12 and 8 and whose altitude is 14. Ex. 1142. Of a frustum of a regular hexagonal pyramid, the sides of whose bases are 10 and 5, and whose altitude is 18. Ex. 1143. Find the volume of a truncated right triangular prism, the sides of whose base are 6, 8, and 10, and whose lateral edges are 5, 8, and 7, respectively. Ex. 1144. The total surface of a cube is 336 cu. in. Find its volume. Ex. 1145. The lateral area of a right prism is 140, and its base is a triangle whose sides are 5, 7, and 8. Find its volume. Ex. 1146. Find the volume of an oblique prism whose base is a regular hexagon inscribed in a circle of radius 5 in., if a lateral edge bé 8 in. and its projection on the plane of the base be 3 in. Ex. 1147. The base of a pyramid whose altitude is 24 is a square of side 10. Find the area of a section 4 in. from the vertex and parallel to the base. Ex. 1148. How far from the vertex, on the lateral edge a, must a plane Il to the base be passed to divide a pyramid into two equivalent parts ? Ex. 1149. The altitude of a pyramid, having a regular hexagon, of side 8 in., for its base, is 28 in. Find the distance from the vertex of a section whose area is 3V3 sa, in. Ex. 1150. Find the surface and volume of a cube in which the diagonal of each face is 15 in. Ex. 1151. Find the surface and volume of a cube whose diagonal is 30 in. Ex. 1152. The right-section of a right prism is a quadrilateral ABCD, in which AB= 7 in., BC = 20 in., CD = 15 in., DA= 24 in., and the angles at A and C are right angles. If the height of the prism is 12 in., find its entire surface and volume. Ex. 1153. Find the lateral area of a right pyramid having the same base and height as a cube whose edge is 10 in. Ex. 1164. The diagonals of a rectangular parallelopiped are equal. 2 Ex. 1155. Find the lateral area of the frustum of a pyramid the alti. tude of which is 15 in., and the bases are squares of sides 40 in. and 24 in., respectively. Ex. 1156. A cone 12 in. in height and 16 in. in diameter at the base is cut by a plane parallel to the base and 9 in. from it. Find the lateral area and volume of the frustum so formed. Ex. 1157. The midpoints of two pairs of opposite edges of a tetraedron determine a parallelogram. Ex. 1158. The length of a prism is 10 in., and a right section is a regular hexagon of side 8 in. Find the surface and volume of the greatest possible cylinder, of the same axis, that can be cut from the prism. Ex. 1159. The base of a right pyramid is a regular hexagon of side 20 in., and the lateral faces are inclined to the base at an angle of 60°. Find the volume. Ex. 1160. Lines joining the midpoints of opposite edges of a tetraedron meet in a point and bisect each other. Ex. 1161. The altitude of a cone is 27 in., and its curved surface is 7 times the area of its base. Find the radius of the base. Ex. 1162. The sum of the squares of the four diagonals of a parallelopiped is equal to the sum of the squares of the twelve edges. Ex. 1163. In any tetraedron the straight lines which join the intersections of the medians of any face with the opposite vertex meet in a point which divides each line in the ratio 1:3. Ex. 1164. Find the edge of the greatest cube that can be cut from a right pyramid a inches high, and having a square base of side b inches, one face of the cube being in the plane of the base of the pyramid. Ex. 1165. Find the total surface of a regular tetraedron, if the perpendicular from one vertex to the opposite face is 5 inches. Ex. 1166. The ends of a frustum of a cone are respectively 8 inches and 2 inches in diameter. If the lateral area is equal to the area of a circle whose radius is 5 inches, find the altitude of the frustum. Ex. 1167. The corners of a cube are cut off by planes which pass through the middle points of each set of edges meeting at a common vertex. If the edge of the cube is 2 feet, find the volume of the solid formed. BOOK VIII THE SPHERE 671. DEF. A sphere is a solid bounded by a surface, all the points of which are equally distant from a point within called the center. 672. DEF. The radius of a sphere is a straight line drawn from the center to any point in the surface. 673. DEF. The diameter of a sphere is a straight line passing through the center and terminated at either end by the surface. 674. From the definitions it follows that (1) All the radii of a sphere are equal, and all diameters are equal. (2) A semicircle rotating about its diameter generates a sphere. (3) Two spheres are equal if their radii are equal, and conversely. (4) A point is without a sphere if its distance from the center is greater than the radius. Ex. 1168. The radii of two spheres are respectively 10 in. and 4 in., their lines of centers (i.e. the line joining their centers) is 7 in. Is every point of the smaller sphere lying within the larger one ? PROPOSITION I. THEOREM 675. Every section of a spliere made by a plane is a circle. Hyp. CBD is the intersection of plane MN, and a sphere whose center is 0. To prove CBD is a circle. OC = OB, (474) Q.E.D. 676. Cor. 1. A circle nearer to the center of a sphere is. greater than one more remote. For since AC = OC’ – AO”, AC is the smaller, the greater AO. 677. DEF. A great circle of a sphere is a section made by a plane passing through the center. |