Longmans' School Mensuration; With an Additional Chapter and Exercises (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1897 Excerpt: ...for the pyramid. For, as the perimeter of a polygon, inscribed in a circle, becomes the circumference of the circle when the number of the sides of the polygon is increased indefinitely, so the cone may be regarded as a pyramid having for its base a polygon of an indefinite number of sides. Lateral surface of pyramid = perimeter of base x 8ant ne'gnt But perimeter of base of polygon)., of indefinite number of sides clrcumference of base of cone.-. Lateral surface of cone = c x--= 2-r x-= rrh 2 2 And total surface = r2 + nrh--mir + h) These are the results previously obtained. (c) To find the slant height, perpendicular height and radios of the cone being given. Let ABC be the cone. AO = perpendicular height = h. BO = radius of base-r. Then AB = slant height. AOB is a right-angled triangle..-. AB2 = OA2 + OB2.-. Slant height-= //7-2 + r2 Bule.--Take the square root of the sum of the squares of the perpendicular height and the radius of the base. (d) To find tho volume of a cone. The volume of a cone may be obtained from that of the pyramid. For it may be considered as a pyramid having an indefinite number of sides. Therefore the perimeter of the pyramid ultimately becomes the circumference of the cone. /. Volume of a cone = area of the base x one-third of the perpendicular height. Let r = radius of base, and h = perpendicular height..-. Volume = irr2 x n O Rule.--Multiply the area of the base by one-third of the perpendicular height. (See Easy Exercises XIII. And XIV.) EXERCISE XXVIII. 1. Find the lateral surface of a cone whose diameter is 17 ft. 2 in., and perpendicular height 21 ft. 2. How many yards of canvas f yd. wide will be required for a conical tent 20 ft. in diameter and 15 ft. high? 3. Find the volume of a cone the radius of whose base ...