XXXV. SEGMENTS OF A RIGHT CIRCULAR CYLINDER. RING. 326. Simple rules can be given for finding the areas of the curved surfaces of certain segments of a right circular cylinder; as we will now shew. 327. Suppose a right circular cylinder cut into two parts by a plane parallel to the axis; the surface of each part consists of two segments of a circle, a rectangle, and another portion which we shall call the curved surface. The area of each of the segments of a circle can be found by Arts. 185 and 186. The area of the rectangle can be found by Art. 134. The area of the curved surface can be found by the following rule: multiply the length of the arc of the base by the height of the cylinder. Or we may find the area of the rectangle and the curved surface together by the Rule of Art. 323, multiply the perimeter of the base by the height of the cylinder. 328. Suppose a solid has been obtained by cutting a right circular cylinder by a plane, inclined to the axis, which does not meet the base of the cylinder. The surface of this solid consists of the base which is a circle, the other end which is also a plane curve, and another portion which we shall call the curved surface. The area of the base can be found by Art. 168. No Rule has been given in this work for finding the area of the other end exactly; but the area might be found approximately by Art. 193: F A G D B we may remark that this plane curve is called an ellipse, and is of great importance in mathematical investigations. The area of the curved surface can be found by the following Rule: multiply the circumference of the base by the height of the solid. 329. The height of the solid in the preceding Rule is to be understood in the same sense as in Art. 256. The truth of the Rule may be shewn in the manner of Art. 257. 330. Suppose a solid has been obtained by cutting a right circular cylinder by two planes inclined to the axis, which do not meet each other. The area of the curved surface will be found by multiplying the circumference of the base of the cylinder by the height of the solid. The height of the solid is to be understood as in Art. 259. The Rule follows from the fact that the solid may be supposed to be the difference of two solids of the kind considered in Art. 256 or Art. 328. 331. To find the area of the surface of a solid ring. RULE. Multiply the circumference of a circular section of the ring by the length of the ring. The length of the ring is to be understood as in Art. 261. The Rule may be illustrated in the manner of Art. 260. 332. Examples: (1) The radius of the circular section of a ring is one inch, and the length of the ring is ten inches. The circumference of the circular section of the ring is 2 × 31416 inches; therefore the area of the surface of the ring in square inches is 10 × 2 × 31416, that is 62.832. Thus the area of the surface is 63 square inches nearly. (2) The inner diameter of a ring is 7 inches, and the outer diameter is 8 inches. As in Art. 262 we find that the radius of the circular section is of an inch, and the length of the ring is 23:562 inches; therefore the area of the surface of the ring in 1 2 square inches=31416 × 23'562=37·01... EXAMPLES. XXXV. 1. Find the area of the curved surface of the smaller of the two pieces in the diagram of Art. 327, supposing the height of the solid to be 4 feet, the radius of the circle 15 inches, and the chord of the circle equal to the radius. 2. Find the area of the curved surface of the smaller of the two pieces in the diagram of Art. 327, supposing the height of the solid to be 4 feet 2 inches, the radius of the circle to be 8 inches, and the chord to subtend a right angle at the centre of the circle. 3. The radius of the base of a cylinder is 16 inches; a piece is cut off by two planes inclined to the axis of the cylinder, which do not meet each other; the length of the portion of the axis between the two planes is 35 inches: find the area of the curved surface. Find in square inches the areas of the surfaces of rings having the following dimensions: 4. Length 20 inches; circumference of cross section 4 inches. 5. Length 25 inches; radius of cross section of an inch. 6. Outer diameter 47 inches; inner diameter 4.1 inches. 7. Inner diameter 11 inches; diameter of cross section 2 inches. 8. Outer diameter 26 inches; diameter of cross section 4 inches. 9. Outer diameter 25 inches; circumference of cross section 10 inches. 10. Inner diameter 20 inches; circumference of cross section 12 inches. 11. The area of the surface of a ring is 100 square inches; the radius of the cross section is 1 inch: find the length of the ring. 12. The area of the surface of a ring is 120 square inches; the length is 20 inches: find the inner diameter. XXXVI. RIGHT CIRCULAR CONE. 333. The surface of a right circular cone consists of a circular base and another portion which we shall call the curved surface. A 334. Let ABCD be a sector of a circle. Cut it out of paper or cardboard. Then let it be bent until the edge AB just comes into contact with the edge AD. It is easy to see that by proper adjustment we can thus obtain a thin shell, the outside of which will correspond to the curved surface of a right circular cone: A becomes the vertex of the cone, AB becomes the slant height of the cone, and BCD becomes the circumference of the base of the cone. Hence it will follow that the curved surface of a B right circular cone is equal to a sector of a circle, the radius of the sector being the slant height of the cone, and the arc of the sector being the circumference of the base of the cone: thus we obtain the Rule which we shall now give. 335. To find the area of the curved surface of a right circular cone. RULE. Multiply the circumference of the base by the slant height of the cone, and half the product will be the area of the curved surface. 336. Examples: (1) The radius of the base of a right circular cone is 8 inches, and the slant height is 14 inches: find the area of the curved surface. The circumference of the base in inches = 2 × 8 × 31416 Thus the area of the curved surface is 352 inches nearly. (2) The radius of the base of a right circular cone is 4 feet, and the height of the cone is 3 feet: find the area of the whole surface. We must first find the slant height of the cone; by Art. 55, the slant height in feet is the square root of 16 +9, that is, the square root of 25, that is 5. 1 × 5 × 2 × 4×3·1416=20 × 3·1416 = 62.832. Thus the area of the curved surface is 62-832 square feet. The area of the base in square feet = 4 x 4 x 31416 = 50'2656. Therefore the area of the whole surface in = square feet 62.832+50°2656=113.0976. 337. The following inferences may be easily drawn from the Rule in Art. 335: If the slant height of the right circular cone be twice the radius of the base the area of the curved surface is twice that of the base of the cone; if the slant height be three times the radius of the base, the area of the curved surface is three times that of the base of the cone; and so on. We may sum up these inferences thus: the slant height of a right circular cone bears the same proportion to the radius of the base as the area of the curved surface bears to the area of the base. 338. We will now solve some exercises. (1) The area of the whole surface of a right circular cone is 24 square feet, and the slant height is twice the radius of the base: find the radius of the base. By Art. 337 the area of the curved surface is equal to twice the area of the base; and thus three times the area of the base is 24 square feet: therefore the area of the base is 8 square feet. We must then find the radius of |