Page 1 : V Academy, 13-Surface Areas and Volumes, Prof. Jagadeesh Alkur, M.Tech (PhD), , FOR ADMISSION, Contact- 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 2 : 10th CBSE Mathematics, , Surface Areas and, Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 3 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 4 : 10th CBSE Mathematics, , Surface Areas and Volumes, , CUBOID, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 5 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area and Volume of Cuboid, A cuboid is the region covered by its six rectangular faces. The surface area of a cuboid is equal to, the sum of the areas of its six rectangular faces., Surface area of the cuboid, Consider a cuboid whose dimensions, are l × b × h, respectively, Cuboid with length l, breadth b and height h, The total surface area of the cuboid (TSA), = Sum of the areas of all its six faces, TSA (cuboid) = 2(l × b) + 2(b × h) + 2(l × h) = 2(lb + bh + lh), , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 6 : 10th CBSE Mathematics, , Surface Areas and Volumes, Lateral surface area (LSA) is the area of all the sides apart from the top and, bottom faces., The lateral surface area of the cuboid = Area of face AEHD + Area of face, BFGC + Area of face ABFE + Area of face DHGC, LSA (cuboid) = 2(b × h) + 2(l × h) = 2h(l + b), Length of diagonal of a cuboid =√(l2 + b2 + h2), , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 7 : 10th CBSE Mathematics, , Surface Areas and Volumes, Volume of a Cuboid, The volume of a cuboid is the space occupied within its six rectangular faces., Volume of a Cuboid = (base area) × height = (lb)h = lbh, Surface Area and Volume of Cube, A cube is a three-dimensional solid, that has six square faces, twelve edges and, eight vertices., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 8 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area of Cube, For a cube, length = breadth = height, Cube with length lTSA (cube) =2 × (3l2) = 6l2, Similarly, the Lateral surface area of cube = 2(l × l + l × l) = 4l2, Note: Diagonal of a cube =√3l, Volume of a Cube, Volume of a cube = base area × height, Since all dimensions of a cube are identical, volume = l3, Where l is the length of the edge of the cube., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 9 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 10 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area and Volume of Cylinder, A cylinder is a solid shape that has two circular bases, connected with each other, through a lateral, surface. Thus, there are three faces, two circular and one lateral, of a cylinder. Based on these, dimensions, we can find the surface area and volume of a cylinder., Surface Area of Cylinder, Take a cylinder of base radius r and height h units. The curved surface of this cylinder, if opened, along the diameter (d = 2r) of the circular base can be transformed into a rectangle of, length 2πr and height h units. Thus,, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 11 : 10th CBSE Mathematics, , Surface Areas and Volumes, Transformation of a Cylinder into a rectangle.CSA of a cylinder of base radius r and, height h = 2π × r × h, TSA of a cylinder of base radius r and height h = 2π × r × h + area of two circular bases, =2π × r × h + 2πr2, =2πr(h + r), Volume of a Cylinder, Volume of a cylinder = Base area × height, = (πr2) × h = πr2h, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 12 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area and Volume of Right Circular Cone, A cone is a 3d shape that has one circular base and narrows smoothly from base to a point,, called vertex., Surface area of cone, Consider a right circular cone with slant length l, radius r and height h., Cone with base radius r and height h, CSA of right circular cone = πrl, TSA = CSA + area of base = πrl + πr2 = πr(l + r), , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 13 : 10th CBSE Mathematics, , Surface Areas and Volumes, Volume of a Right Circular Cone, The volume of a Right circular cone is 1/3 times that of a cylinder of the, same height and base., In other words, 3 cones make a cylinder of the same height and base., The volume of a Right circular cone =(1/3)πr2h, Where ‘r’ is the radius of the base and ‘h’ is the height of the cone., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 14 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area and Volume of Sphere, A sphere is a solid that is round in shape and the points on its surface are at, equidistant from the center., Surface area of Sphere, For a sphere of radius r, Curved Surface Area (CSA) = Total Surface Area (TSA) = 4πr2, Volume of Sphere, The volume of a sphere of radius r = (4/3)πr3, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 15 : 10th CBSE Mathematics, , Surface Areas and Volumes, Surface Area and Volume of Hemisphere, A hemisphere is shape that is half of the sphere and has one flat surface. The other side of, the hemisphere is shaped as a circular bowl. See the figure below., Surface Area of Hemisphere, We know that the CSA of a sphere = 4πr2., A hemisphere is half of a sphere., CSA of a hemisphere of radius r = 2πr2, Total Surface Area = curved surface area + area of the base circle, TSA = 3πr2, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 16 : 10th CBSE Mathematics, , Surface Areas and Volumes, Volume of Hemisphere, The volume (V) of a hemisphere will be half of that of a sphere., The volume of the hemisphere of radius r = (2/3)πr3, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 17 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 18 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 19 : 10th CBSE Mathematics, , Surface Areas and Volumes, EXERCISE 13.1, , 1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the, resulting cuboid., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 20 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 21 : 10th CBSE Mathematics, , Surface Areas and Volumes, , A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere, of same radius. The total height of the toy is 15.5 cm. Find the total, surface area of the toy., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 22 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 23 : 10th CBSE Mathematics, , Surface Areas and Volumes, A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends, (see Fig. 13.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5, mm. Find its surface area., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 24 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 25 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 26 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 27 : 10th CBSE Mathematics, , Surface Areas and Volumes, A tent is in the shape of a cylinder surmounted by a conical top. If the height and, diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of, the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the, cost of the canvas of the tent at the rate of 500 per m2., (Note that the base of the tent will not be covered with canvas.), , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 28 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 29 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 30 : 10th CBSE Mathematics, , Surface Areas and Volumes, A wooden article was made by scooping out a hemisphere from each end of a solid, cylinder, as shown in Fig. 13.11. If the height of the cylinder is 10 cm, and its base is, of radius 3.5 cm, find the total surface area of the article., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 31 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 32 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 33 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 34 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 35 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 36 : 10th CBSE Mathematics, , Surface Areas and Volumes, , Volume of a Combination of Solids, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 37 : 10th CBSE Mathematics, , Surface Areas and Volumes, EXERCISE 13.2, , A solid is in the shape of a cone standing on a hemisphere with both their radii being, equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the, solid in terms of π., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 38 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 39 : 10th CBSE Mathematics, , Surface Areas and Volumes, A gulab jamun, contains sugar syrup up to about 30% of its volume. Find, approximately how much syrup would be found in 45 gulab jamuns, each shaped like, a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 40 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 41 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 42 : 10th CBSE Mathematics, , Surface Areas and Volumes, A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top,, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of, which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water, flows out. Find the number of lead shots dropped in the vessel., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 43 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 44 : 10th CBSE Mathematics, , Surface Areas and Volumes, A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing, on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of, water such that it touches the bottom. Find the volume of water left in the cylinder, if, the radius of the cylinder is 60 cm and its height is 180 cm., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 45 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 46 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 47 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 48 : 10th CBSE Mathematics, , Surface Areas and Volumes, , Conversion of Solid from One Shape to Another, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 49 : 10th CBSE Mathematics, , Surface Areas and Volumes, EXERCISE 13.3, , Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single, solid sphere. Find the radius of the resulting sphere., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 50 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 51 : 10th CBSE Mathematics, , Surface Areas and Volumes, , A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread, out to form a platform 22 m by 14 m. Find the height of the platform., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 52 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 53 : 10th CBSE Mathematics, , Surface Areas and Volumes, A container shaped like a right circular cylinder having diameter 12 cm and height 15, cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and, diameter 6 cm, having a hemispherical shape on the top. Find the number of such, cones which can be filled with ice cream., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 54 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 55 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 56 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 57 : 10th CBSE Mathematics, , Surface Areas and Volumes, How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to, form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 58 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 59 : 10th CBSE Mathematics, , Surface Areas and Volumes, Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How, much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 60 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 61 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 62 : 10th CBSE Mathematics, , Surface Areas and Volumes, A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm, is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter, 6 cm, having a hemispherical shape on the top. Find the number of such cones which, can be filled with ice cream., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 63 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 64 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 65 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 66 : 10th CBSE Mathematics, , Surface Areas and Volumes, , Frustum of a Cone, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 67 : 10th CBSE Mathematics, , Surface Areas and Volumes, A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters, of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 68 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 69 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 70 : 10th CBSE Mathematics, , Introduction to Syllabus, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 71 : 10th CBSE Mathematics, , Surface Areas and Volumes, EXERCISE 13.4, A fez, the cap used by the Turks, is shaped like the frustum of a cone (see Fig. 13.24). If, its radius on the open side is 10 cm, radius at the upper base is 4 cm and its slant height, is 15 cm, find the area of material used for making it., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 72 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 73 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 74 : 10th CBSE Mathematics, , Surface Areas and Volumes, A container, opened from the top and made up of a metal sheet, is in the form of a, frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20, cm, respectively. Find the cost of the milk which can completely fill the container, at the, rate of Rs. 20 per litre. Also find the cost of metal sheet used to make the container, if it, costs Rs. 8 per 100 cm2., , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 75 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 76 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 77 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 78 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 79 : 10th CBSE Mathematics, , Surface Areas and Volumes, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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Page 80 : 10th CBSE Mathematics, , V Academy, , Contact - +91 81 47 37 58 73, 99 0 28 4 777 4, mail-
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