628. DEF. A prism is circumscribed about a cylinder when its lateral edges are parallel to the elements of the cylinder and its bases are circumscribed about the bases of the cylinder. 629. DEF. A section of a cylinder is the figure formed when the cylinder is intersected by a plane; a right section is a section formed by a plane perpendicular to the elements. PROPOSITION XXIV. THEOREM 630. Every section of a cylinder made by a plane passing through an element is a parallelogram. Hyp. ABCD is a section of cylinder AC, made by plane through element AB. To prove ABCD is a parallelogram. Proof. Any straight line through D in plane AC I to AB is an element of the cylindrical surface. (Why ?) Since this line is in the plane AC and is an element of the cylindrical surface, it must be their intersection, and therefore coincides with DC. Also .. DC is a straight line | to AB. (Why?) Q.E.D. 631. COR. Every section of a right cylinder made by a plane passing through an element is a rectangle. Ex. 1127. The altitude of a right cylinder is 12 inches and the radius of the base is 6 inches. Find the area of a section made by a plane passing through an element and perpendicular to a radius at a point whose distance from the center measured on this radius is Ex. 1128. What is the area of the figure formed in the preceding exercise when the plane passes through the center of the base? PROPOSITION XXV. THEOREM 632. The bases of a cylinder are equal. Hyp. ABE and DCG are the bases of the cylinder BG. To prove Base ABE = base DCG. Proof. A, B, E, are any three points in lower base of cylinder, AD, BC, and EG are elements. Draw AE, AB, BE, DG, DC, and CG. The figures AC, BG, and ED are parallelograms. (Why?) AB = DC, BE = .. A ABE = A DGC. Place the lower base on the upper base so that the equal A coincide. Then the bases will coincide, because the points A, B, and E are any three points in the perimeter of the lower base, and therefore every point in the perimeter of the lower base will fall in the perimeter of the upper base. ..the bases are equal. Q.E.D. 633. COR. 1. Any two parallel sections cutting all the elements of a cylinder are equal. HINT.(What is the solid included between these sections ?) 634. COR. 2. Any section of a cylinder parallel to the base is equal to the base. PROPOSITION XXVI. THEOREM 635. The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by an element. Hyp. S is the lateral area, P the perimeter of a right section, and E an element of the cylinder AK; S' is the lateral area, P' the perimeter of a section of a prism with a regular polygon as base, inscribed in cylinder AK. Proof. The edge of the inscribed prism coincides with an element of the cylinder. .. S'P' x E. (Why?) (547) If the number of faces of the inscribed prism be increased, S' will approach S as a limit, and P' will approach P as a limit. But S'P' x E. .. SPX E. (213) 636. COR. 1. The lateral area of a cylinder of revolution is the product of the circumference of its base by its altitude. 637. COR. 2. If S denote the lateral area, T the total area, H the altitude, and R the radius of a cylinder of revolution, S = 2 πRH, and T=2 TR (H+ R). PROPOSITION XXVII. THEOREM 638. The volume of a circular cylinder is equal to the product of its base by its altitude. K Hyp. V the volume, B the base, and H the altitude, of cylinder AK; V' the volume, B' the regular polygon forming the base of a prism inscribed in AK. R, HINT. Use the Theorem of Limits as in preceding proposition. 639. COR. For a cylinder of revolution with radius of base V = TR2 × H. Ex. 1129. Two cylinders of revolution have equal altitudes and their radii are respectively 3 and 4. Find a third cylinder of revolution of the same altitude and equivalent to the sum of the two given cylinders. |