AREA OF PLANE FIGURES 1. Triangle Area of triangle = 1 x base x height Area of an equilateral triangle = 4 ( side) If the length of the sides of a triangle are a, b and c and if the semiperimeter s = ( a b c) Area of the triangle = s( s a) ( s b)( s c). Parallelogram Area = base x height. Rectangle Area= Length x Breadth Perimeter = Sum of all sides = (Length + Breadth) 4. Square Let `a' be the side of square Diagonal = a 1 Area = a = diagonal Perimeter = 4a 5. Trapezium then, Area = 1 6. Rhombus (sum of parallel sides) x (distance between them) Area = 1 (Product of the diagonals) 7. Circle Let r be the radius of the circle Area of a circle = r, or 14. 7 Circumference of a circle = r Length of an Arc = r 60 Area of a sector = r 60 The unit of area is square metre Volume & area of Solid Structures. 1. Cuboid Let l, b, and h are respectively the length, breadth and height of a cuboid Volume of a cuboid = lbh Longest diagonal of a cuboid = l b h Total surface area = (lb+bh+lh) Area of a room = (length x breadth)
Area of 4 walls of a room. Cube = (l+b) height Let `a' be the side of a cube Volume of the cube = a Longest diagonal = x (a Total surface area = 6a Total length of all edges = 1a.. Cylinder Volume = r h; r - radius of the base Base area = r h - height Curved surface area = r h Total surface area = rh r 4. Pyramid Volume = 1 x (area of base) x height 5. Cone If the radius of the base of a cone is r, the height is h and the slant height is l then l = h + r Volume = 1 r h Area of curved surface = rl Total surface area = rl + r 6. Sphere Volume = 4 = r(l + r) r ; r- radius of the sphere Curved surface area = 4 r 7. Hemisphere Volume = r ; Curved surface area = Total surface area = r r Volume of spherical shell = 4 ( R r ) Volume of a metal in a hollow pipe = ( R r ) h; R - external radius Total surface area of an open pipe = Rh rh ( R r ) The unit of volume is cubic metre. Solved Examples
1. Find the area of a right triangle whose hypotenuse is 5 metres and base is metres. Ans: Altitude of the triangle Area of the triangle = 5 16 4m. = 1 x base x altitude = 1 x x 4 6 sq. m.. A lawn is in the form of a rectangle having its sides in the ratio :. The area of the lawn is 1 6 length and breadth of the lawn. Ans: Let the length and bread be xand xmetres. 1 Then 6x 6 1 x 10000 x 50 Length = x 50 50m hectare. Find the Breadth = x 50 1 m. A rectangular grassy plot is 11 metres by 78 metres. It has a gravel path.5 metres wide all round it on the inside. Find the area of the path and the cost of constructing it at 7 paise per 1000 sq.cm. Ans: Area of path = Area of grassy plot - Area of plot excluding the path = (11 x 78-107 x 7) = 95 m.5 78.5 Cost on 1000 sq.cm. = 7 paise. Total Cost 11 = Rs. 7 1000 95x100 x100 x Rs. 6, 660 100 4. How many bricks will be required for a wall 8m long, 6 m high and.5 cm thick if each brick measures 5 cm by 11.5 cm by 6 cm? Ans: Volume of wall = (800x600x.5) cu.cm Volume of brick = (5x11.5x6) cu.cm. Number of bricks = Volumeof thew all Volume of abrick 800x600x. 5 = 6400 5x115. x6 5. Half cubic metre of gold sheet is extended by hammering so as to cover an area of 1 hectare. Find the thickness of the gold sheet.
Ans: Volume of sheet = 1 cu.metre = 1 x 100x100 x100 cu.cm Area of sheet=1hectare =10,000 sq. metre = [10,000x100x100] sq.cm. Thickness = Volume Area 1x100 x100 x100 = x10, 000x100 x100 1 = cm = 0.005 cm. 00 6. A rectangular lawn 60 metres by 40 metres has two roads each 5 metres wide, running in the middle of it, one parallel to length and the other parallel to the breadth. Find the cost of gravelling them at 60 paise per square metre. Ans: Area of the road ABCD=60x5=00 sq.m Area of the road EFGH = 40x5 = 00 sq.m. E H A M L D B J K C F G Area of JKLM = 5x5 = 5 sq.m. Area of the path of be gravelled = (00+00-5) = 475 sq.m. = cost of gravelling the path. 60x475 = Rs. Rs. 100 Quicker Method 85 Area of the road = (width of the road) x (length + breadth of the lawn - width of the road) Cost = = 5(60 + 40-5) = 475 sq.m 60x475 Rs. 85 100 7. A rectangular courtyard.78 metres long and 5.5 metres broad is to be paved exactly with square tiles, all of the same size. What is the largest size of such a tile? Also find the number of tiles? Ans: Largest size of tile =H.C.F. of 78 cm and 55 cm=1 cm. Area of courtyard = (78x55) sq.cm. 78x55 Number of tiles = 1x 1 450 8. The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must be the wheel make in order to keep a speed of 66 km. per hour? Ans: Distance covered by wheel in 1 minute
66x1000, x100 = = 1,10,000 cms. 60 Circumference of wheel = x x 70 440 cm. 7 Number of revolutions in 1 minute 110,, 000 = 50 440 9. The length of a rectangle is increased by 60%. What per cent should the width be decreased to maintain the same area? Ans: Let the length of rectangle be increased by x%. Then the decrease percentage of breadth to maintain the same area 100 = x 100 x 100 In the above problem decrease percentage of breadth = 60 7 1 100 60 % 10. A cube of sides cm. is melted and smaller cubes of sides 1 cm. each are formed. How many such cubes are possible? Ans: Quicker method Required Number Original length of side = New length of side = 1 7 11. Three cubes of metals whose edges are, 4 and 5 cm. respectively are melted and formed into a single cube. If there is no loss of metal in the process, find the side of the new cube. Ans: Quicker method When many cubes integrate into one cube, the side of the new cube is given by side = Here, side sum of cubes of sides of all thecubes = 4 5 7 64 15 6cm 1. If the radius of a cylinder is doubled and the height is halved, what is the ratio between the new volume and previous volume? Ans: Let the initial radius and height of the cylinder be r cm and h cm respectively. Then V r h 1 V x r x h r h ( ) New Volume r h Pr evious volume r h 1 Required ratio = :1
PRACTICE TEST 1. The perimeter of a rectangle is 8 m. and its length is 5m. The breadth of the rectangle is: a) 14 m. b) 16m c) 18 m. d) 1 m.. The length of a plot of land is 4 times its breadth. A play ground measuring 100m occupies one third of the total area of the plot. What is the length of the plot, in metres? a) 90 b) 80 c) 60 d) 10. If the width of a rectangle is m. less than its length and its perimeter is m., the area of the rectangle is: a) 4m b) 108m c) 99m d) 6m 4. The sides of a rectangular park are in the ratio : and its area is 750m. The cost of fencing it at 50 paise per metre is: a) Rs. 1.50 b) Rs. 75 c) Rs. 187.50 d) Rs. 15 5. The least number of equal square slabs that can be fitted in a verandah 10.5m. long and m wide is a) 15 b) 14 c) 1 d) 1 6. Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the new cuboid to that of the sum of the surface area of the cubes. a) 7:9 b) 6:5 c) 7:8 d) 5:6 7. If the radii of the two sphere are in the ratio 1:4, what will be the ratio of their volumes? a) 1:8 b) 1:16 c) 1:4 d) 1:64 8. Six spherical copper balls of radius `r' are melted and cast into a cylinderical rod of the metal of same radius. The height of the rod will be a) 4r b) 6r c) 8r d) 1r 9. A man walked 0 metres to cross a rectangular field diagonally. If the length of the field is 16 metres, what is the breadth of the field? a) 16 b) 4 c) 1 d) 8 10. If the ratios of areas of two squares is 9:1, the ratio of their perimeter is a) 9:1 b) :4 c) :1 d) 1: 11. If the area of a cirlce is equal to its circumference, then diameter of the circle is: a) cms. b) 4cms. c) 1 cm. d) cms 1. If the side of a sqaure is increased by 0%, the area of the square is increased by: a) 40% b) 5% c) 44% d) 0% 1. If the area of a circle is decreased by 6% then radius of a circle decreases by a) 0% b) 18% c) 6% d) 64% 14. If the area of a square is increased by 96%, then the side of the square is increased by a) 96% b) 48.5% c) 40% d) 49% 15. A rectangular hall 10m. long and 8m. broad, is surrounded by a verandah metres wide. Find the area of the verandah. a) 104m b) 88m c) 96m d) 10m
16. The surface area of a cube is 600m. The length of its diagonal is: a) 10 10 cm b) cm c) 10 cm d) 10 cm. 17. Two circular cylinders of equal volume have heights in the ratio 1:. The ratio of their radii is: a) 1: b) : 1 c) 1:4 d) :1 18. A copper sphere of radius cm. is beaten and drawn into a wire of diameter 0.cm. The length of the wire is a) 9m. b) 1m. c) 18m. d) 6m. 19. The base of a prism is a triangle of sides m., 4m, and 5m. respectively. The height of the prism is 10m. Then its volume is: a) 60m b) 75m c) 4m d) 8m 0. One cubic metre of aluminium sheet is extended by hammering, so as to cover the roof with an area 10,000 square metre. The thickness of the sheet is: a) 0.1cm. b) 0.01m. c).01cm d) 0.5cm ANSWERS TO PRACTICE TEST 1. (b). (d). (d) 4. (d) 5. (b) 6. (a ) 7. (d) 8.(c) 9. (c) 10. (c) 11. (b) 1. (c) 1. (a) 14. (c) 15. (b) 16.(c) 17. (a) 18. (d) 19. (a) 0. (c)