When a sector of 60 degrees is used, what will be the diameter and the slant height, the radius of the sector being 3 inches? (Make no allowance for overlapping.) 12. If you wish to make a hollow paper cone whose slant height shall be 5 inches, and the diameter of whose base shall be 3 inches, how many degrees should the arc of the sector contain? 13. Draw the development of a right pyramid 4 inches in altitude, whose base is a rectangle 3 inches by 2 inches. Is the altitude (slant height) of each of the four triangles the same? 14. Calculate the slant height of each convex face of a rectangular pyramid whose altitude is 12 inches, and whose base measures 10 inches by 18 inches. Find the convex surface. 1284. Surface of Frustum of Pyramid and Cone. When the upper part of a pyramid or of a cone is cut off by a plane parallel to the base, the remaining part is called a frustum. 15. Draw one face of the frustum of a square pyramid. Of the frustum of a triangular pyramid. What figure have you drawn? Calculate its surface when the length of the upper side is 4 inches, that of the lower side is 8 inches, and the slant height of the frustum is 10 inches. 16. Draw the developed convex surface of the frustum of a regular triangular pyramid, each side of the upper base measuring 1 inch, of the lower base 2 inches, the slant height being 2 inches. SUGGESTION. - Locate the apex of the whole pyramid of which the given frustum forms a part. 17. Find the convex surface of the frustum of a square pyramid, one side of the upper base measuring 2 feet, of the lower base 3 feet, and having a slant height of 4 feet. Find the entire surface. 18. Show that the convex surface of the frustum of a pyramid is equal to one-half the sum of the perimeters of the upper and the lower bases multiplied by the slant height. 19. Draw the pattern of a small shade for a candle. Make the upper opening 14 inches in diameter, the lower one 2 inches in diameter, and the slant height 2 inches. 20. How many square inches of tin will be required to make a pan, its upper base being 9 inches in diameter, the lower base 6 inches in diameter, and the slant height 4 inches? (Do not forget the bottom of the pan.) 1285. The frustum of a cone may be considered the frustum of a pyramid whose bases contain a very great number of sides. The convex surface of the frustum of a cone may, therefore, be found by multiplying the half sum of the circumference of the two bases by the slant height. 21. Find the convex surface of a frustum of a cone, the circumferences of the bases being 15 inches and 20 inches, respectively, and the slant height 10 inches. 22. How many square yards are there in the entire surface of a frustum of a cone, the radius of the upper base (r) being 3 yards, of the lower base (R) 5 yards, and the slant height 6 yards? Circumference of upper base 2 πг; of lower base = 2πR. = Convex surface = (2πг + 2 πR) X Surface of upper base = π× what? Multiply only once by 3.1416. 23. The diameter, AB, of the upper base of the frustum of a cone measures 6 feet, CD measures 8 feet, the slant height AC measures 9 feet. Find the slant height EA of the part cut from the cone C in making the frustum. E АВ 24. Find the convex surface of the whole cone, ECD, and the convex surface of the part cut off, EAB. CG, CK, CD, CF, CA, and CI are radii; AD and FG are diameters G 1287. If a sphere be cut through at any part, the cut surface will be a circle. When the cutting plane passes through the center of the sphere, the circle is called a great circle; other circles are called small circles. FXGC is a great circle; HYIB and JLEZ are small circles. 25. Find the length of an arc of 60° of a great circle of a sphere whose circumference is 25,000 miles. 26. Calling the arc AĨ in the preceding figure, 30°, the angle BCI will measure 30°. Calculate the radius BI of the small circle when the radius CI of the large circle is 4,000 miles. (IAH arc of 60°; IH chord of 60°.) = 27. If I is 60° from G, a point on the equator, find the length of the circumference of the small circle HYI, assuming the circumference of a great circle to be 25,000 miles. 28. What is the ratio between the length of a degree on the small circle HYI, and the length of a degree of a great circle? 29. Calculate the radius of a small circle formed by passing a plane parallel to GCXF through a point on GA 45 degrees from G. 1288. Surface of Sphere. We have seen (Art. 1151) that it may be experimentally shown that the surface of a sphere is equal to the surface of four of its great circles. Calling the radius of the sphere R, its surface is 4 TR2. 30. Find the surface of a sphere whose diameter is 6 inches. 31. How many square inches are there in the convex surface of a hemisphere whose radius is 3 inches? What is the area of the great circle that forms the base of the hemisphere? Find the entire surface of the hemisphere. 32. Is there any difference between the convex surface of a sphere and its entire surface? Why? VOLUMES. 1289. Prisms and Cylinders. 1. How many one-inch cubes will cover the base of a box 4 inches by 3 inches? If the box is 2 inches deep, how many oneinch cubes will it contain? How many cubic inches are there in the volume of a right prism whose base is a rectangle measuring 4 inches by 3 inches, and whose altitude is 6 inches? 2. If the above hollow prism were divided into two equal parts by a thin partition extending from a vertical edge to one diagonally opposite, how many cubic inches of sand would each part contain? 3. How many cubic inches are there in the volume of a prism whose base is a right-angled triangle 3 by 4 by 5 inches, and whose altitude is 6 inches? 4. Find the volume of a triangular prism, the area of the base being 6 square inches, and the altitude 6 inches. Find the volume of a triangular prism, each side of whose base measures 6 inches, its altitude being 8 inches. 5. What are the solid contents of a pentagonal prism formed by fastening together three triangular prisms whose bases contain, respectively, 12, 16, and 18 square inches, the altitude of each being 15 inches? 6. If a very great number of triangular prisms of the same height are united so as to form a cylinder whose base contains 12.5664 square inches, and whose altitude measures 5 inches, what are the solid contents? 1290. Pyramids and Cones. With a center at C, and a convenient radius, describe an arc AB. Mark off four equal portions v, w, x, and y; and draw the equal chords. Cutting out CAvwxy, with an additional narrow strip along Cy for gumming, and creasing along the lines Cv, Cw, Cx, and Cy, we can fold the paper into a square pyramid. Measure its altitude, and make a square prism of equal altitude and with an equal base. Filling the pyramid with sand, and pouring the sand into the prism, it will be found A 2 B Y that the latter will contain the contents of the former three times; that is, the volume of a square pyramid is one-third that of a square prism having an equal base and an equal altitude. The same ratio will be found true in the case of a triangular, or any other pyramid, as compared with the corresponding prism, and of the cone as compared with a cylinder. 1291. The volume of a pyramid or of a cone is equal, therefore, to the area of the base multiplied by one-third of the altitude. |