121. COROLLARY. The volume of a triangular pyramid is equal to one-third the product of its base and altitude. For, the volume of the prism DEO-ACB equals the product of its base and altitude (82); and from (120), the pyramid ABC-O, constructed with the same base and altitude as the prism, has a volume one-third as great. Hence, the volume of a triangular pyramid is equal to one-third the product of its base and altitude. B DEFINITIONS. 122. A conical surface is a curved surface generated by a moving line passing through a fixed point and continually touching a guiding curve, the curve and the fixed point not lying in the same plane. The generating line in any position is termed an element of the surface. The point in which the elements meet is called the vertex. Ο B It follows from the definition that the conical surface will consist of two parts, each of which has its vertex at O. We shall, however, refer to only the one part, O-ABC, as the conical surface 123. A cone is a solid bounded by a conical surface and a plane called the base passing through all the elements of the surface. The line from the vertex to the centre of the base is called the axis. If the base is a circle, the cone is said to be circular; and if the axis is perpendicular to the circle, the cone is termed a right circular cone. Pi 124. A right circular cone is called a cone of revolution; for if a right triangle be revolved about one of its sides as an axis it will generate a right circular cone. 125. If two similar triangles are revolved about homologous sides as axes, two similar cones of revolution will be generated. 126. A pyramid is inscribed in a cone when the base of the pyramid is inscribed in the base of the cone and the two solids have a common vertex. The lateral edges of the pyramid are elements of the surface of the cone. 127. A cone may be regarded as the limit of an inscribed pyramid when the number of its lateral faces is increased indefinitely. B By indefinitely increasing the number of lateral faces of the pyramid O-ABCD inscribed in the cone O-ABCD, the pyramid is made to approach the cone as its limit. Ultimately, then, the pyramid in its entirety becomes the cone. For definitions of altitude, right section, frustum, etc., of a cone, see corresponding definitions under pyramid. 128. From the above definitions we may readily deduce the following: In the right circular cone: 129. All elements of the surface are equal. 130. Sections embracing the axis are isosceles triangles. 131. Right sections are circles. 132. The circumference of the right section bisecting the altitude equals one-half the circumference of the base. In the frustum of the right cone: 133. All elements of the surface are equal. 134. Sections embracing the axis are trapezoids. 135. The circumference of the right section bisecting the altitude equals one-half the sum of the circumferences of the bases. The demonstration of the above is left to the student. 136. An element of the surface is taken as the slant height (129 and 133). PROPOSITION XXIII. 137. THEOREM. The volume of any pyramid is equal to one-third the product of its base and altitude. = Let V represent the volume of a pyramid, BCDEF the base, and Ax the altitude; then will V (BCDEF) (Ax). } Divide the pyramid into triangular pyramids by the planes A'CE and A'CF. Represent the volumes of these triangular pyramids by V', V", etc. Now, V' = } (BCF) (Ax) . . . (121); and V" (FCE) (Ax), etc. = Hence, V'+V" + V""' = } (BCF + FCE + ECD) (Ax). But, V' + V'' + V""' = V, and, BCF+FCE + ECD = BCDEF. Hence, V (BCDEF) (Ax). = } Q. E. D. 138. COROLLARY. The volume of any cone is equal to one-third the product of the base and altitude. For, in any cone A'-BCDEF inscribe a pyramid A'-BCDEF. By indefinitely increasing the number of its lateral faces the inscribed pyramid is made to approach the cone as a limit (127). Hence, it is true of the cone as of the pyramid, that the volume is equal to one-third the product of the base and altitude. FRUSTUMS OF PYRAMIDS AND CONES. 139. PROBLEM. Given the bases and altitude of the frustum of a pyramid to find the altitude of the smaller pyramid removed to form the frustum. Let B represent the area of the lower base. Then H+ pyramid. We are to find the value of x in terms of B, b, and H. 140. From the conditions given in (139), to find the volume of the entire pyramid of which the frustum is a part. Let V represent the required volume, then we are to find. the value of V in terms of B, b, and H. 141. From the conditions given in (139), to find the volume of the frustum. Let V' represent the volume of the smaller pyramid. We are to find the value of F in terms of B, b, and H. |