357 Applied Math Mensuration of Cone Chapter 18 Mensuration of Cone 18.1 Cone: A cone is a solid generated by a line, one end of which is fixed and the other end describes a closed curve in a plane. The fixed point is called the vertex or apex. Axis: The axis of the cone is the distance from the apex to the centre of the base. Altitude: It is the perpendicular distance from the apex to the base of the cone. It is also called height of the cone. Circular Cone: A cone whose base is a circle and whose lateral surface tapers uniformly to the apex, is called a circular cone. Right Circular Cone: It is a cone whose base is a circle and whose axis is perpendicular to the base. A right circular cone can also be described when a right triangle is rotated about one of its side containing the right triangle. Slant Height: In a right cone it is the distance from the vertex to the circumference of the base. It is denoted by . Note: A cone may also be defined as, when the number of sides of the base of a pyramid is increased indefinitely and the magnitude of each side diminished the surface of the pyramid tends to become the surface of a cone. Hence a cone is a limiting case of a pyramid. 18.2 Volume and Surface area of a Right Circular Cone: If r is the radius of the base, h is the height and is the slant height, then since the cone is the limiting case of a pyramid, therefore volume and surface area of the cone is calculated by the same formula of pyramid. i.e., 1 (i) Volume of the cone = area of the base x height 3 1 = π r2 h 3 358 Applied Math Mensuration of Cone (ii) Lateral surface of a Cone: It is the curved surface area. If a hollow cone is made a cut along a straight line from the vertex to the circumference of the base, the cone is opened out and a sector of a circle with radius is produced. Since, the circumference of the base of the cone is 2π r , therefore the arc length of the sector of the circle is . Thus, Lateral surface area of a cone = area of the Sector OAB 1 = radius x arc length 2 1 = x 2π r 2 = π r Where = slant height = r22 + h Or Lateral surface = 1 Perimeter of the base x slant height 2 = 2π r x = π r (ii) Total surface area = Lateral surface area + area of the base = π r + π r2 = π r( + r) Example 1: Find the volume and the total surface area of a cone of radius 6.6cm and height of 12.5cm. Solution: Here 4 = 6.6cm, h = 12.5cm Volume = 1 = π(6.6)2 x 12.5 3 = 570.199 cu. cm Since 2 h 2 r 2 2 2 = 12.5 + 6.6 = 156.25 +1 43.562 2 π r h = 199.81 3 359 Applied Math Mensuration of Cone = 14.14cm Total surface area = π r( + r) Total surface area = π 6.6(14.14 + 6.60) = 6.6(20.74) = 430.03 sq. cm. Example 2: A pyramid on a regular hexagonal base is trimmed just enough to 1 reduce it to a cone. Show that of the original volume of the 10 pyramid is removed. Solution: Volume of the pyramid = area of the base x height 2 180 Vp = n r tan x h n o 2 = 6 tan 30 r h 1 2 Vp = 2 r h 3 3 r2 = V 2 h p Volume of the cone = Vc = area of the base x height = π r2 h 3 = π hV 2 h p 3 π = V 6 p Vc = 0.9 Vp Volume removed by trimming = Vp – Vc = V1p – 0.9Vp = (1 – 0.9)Vp = 0.1V3 p = Vp 360 Applied Math Mensuration of Cone = volume of the pyramid Example 3: π 3 Find what length of canvas m. wide is required to make a conical 4 tent 8m in diameter and 3m high. Solution: 1 Here, d = 8m, r = 4m, h = 3m Lateral surface of the conical tent10 = π r = π r r22 h where = rh22 = π x 4 16 9 = 20 = 62.83 sq. m Now, area of the tent = 62.83 Width = m Length = = ? l 3 Width x length = area = 62.83 4 4 62.83 x 4 = Fig.18.5 3 = 83.78 m. 361 Applied Math Mensuration of Cone Exercise 18(A) Q.1 The radius of the base of a right circular cone is 6m and the slant height is 6.5m, find the volume and the lateral surface area. Q.2 The generating line of a right circular cone is inclined at an angle of 60o to the horizontal. If the height of the cone be 15cm find its lateral surface and volume. Q.3 The slant height of a cone is 25cm, and the area of its curved surface is 550 sq. cm. find its volume. Q.4 Find the cost of the canvas for 50 conical tents, the height of each being 45cm and the diameter of the base 36cm at the rate of Rs. 9.40 sq. m. Q.5 Find the cost of canvas 3/5m wide at the rate of Rs. 5 per meter, required to make a conical tent, 5m diameter and 2m high, taking no account of waste. Q.6 Find the cost of painting at the rate of Rs. 6.25 per sq. cm. a conical spire 60cm in circumference at the base and 105cm in slant height. Q.7 Find the curve surface area, the whole surface area, the volume and the vertical angle of a cone whose radius of the base is 45cm and height is 48cm. Q.8 A right angled triangle of which the sides are 3cm and 4cm in length is made to turn round the longer side. Find the volume of the cone thus formed. Q.9 A right angled triangle of which the sides are 65, 60 and 25cm in length is made to turn round its hypotenuse. Find the volume of the double cone thus formed. Q.10 The area of the whole surface of a cone is 64 sq. m. and the slant height is 5 times the radius of the base. Find the radius of the base. Q.11 Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 2.8m. Q.12 A cylindrical column 4m wide and 6m high is surmounted (cover) by a cone of the same width and 3m height. Find the area of the sheet metal required to cover its lateral surface. Q.13 The vertical height of a conical tent is 42 dm and diameter of its base is 5.4m. Find the number of persons it can accommodate if each person is to allowed 2916 cu. dm of space. Answers Q.1 94.286 cu.m., 122.52 sq. m. Q.2 471.2 sq. cm., 1178.03 cu. cm Q.3 1232 cu. cm. Q.4 Rs. 128.8 Q.5 Rs. 209.54 Q.6 Rs. 19687.50 362 Applied Math Mensuration of Cone Q.7 9301.55 sq. cm ,15663.27 sq. cm, 37.699 cu. cm, 86o 18 Q.8 37.699 cu. cm, Q.9 36283.97 cu. cm Q.10 1.84m Q.11 5.747 cu. m Q.12 98.05 sq. m Q.13 11 18.3 Frustum of Cone: When a cone is cut through by a plane parallel to its base, the portion of the cone between the cutting plane and the base called a frustum of the cone. The frustum of a cone is the limiting form of the frustum of a pyramid. 18.4 Volume and Surface area of the Frustum of a Cone: Let R and r are radii of the base and top respectively, h is the thickness or height and is the slant height of the frustum of a right cone. Since the frustum of a cone is the limiting form of the frustum of a pyramid. Therefore volume and surface area of the frustum of a cone is calculated by the same formula of the frustum of a pyramid. h (i)1 Volume of the frustum of a cone = (A +A + A A ) 3 1 2 1 2 2 h = (πR2 πr 2 (πR 2 )(πr 2 )) 3 h = (πR22 πr π R r) 3 π h = (R22 r R r) 3 (ii) Lateral surface area = curved surface area Fig.18.6 = (sum of the perimeters of base and top) slant height = (2πR 2π r) = π(R r) (iii) Total surface area = Lateral surface area + area of the base and top = π(R r) + π R22 π r Note: If r, R and h are given, then 2h 2 (R r) 2 Example 5: 363 Applied Math Mensuration of Cone Find the curved and total surface area and the volume of the frustum of a cone whose top and bottom diameters are 6m and 10m and the height is 12m.π Solution: Here d = 6m r = 3m D = 10m R = 5m h = 12m Volume = π x 12 = (25 9 15) 3 = 4(49) = 615.75 cu. cm Lateral surface= π (R r) = 12.166(8) = 305.75 sq. m. In the right triangle ABC 2h 2 (R r) 2 = 144 + 4 Fig.18.7 2 = 148 = 12.166m Total surface area = Lateral surface area + area of the base and top 22 π h = 305.75 + π R π r (R 22 r= 305.75 R r) + 75.54 + 28.27 3 = 412.56 sq.