CHAPTER 11 AREAS OF PLANE FIGURES

Transcription

1 CHAPTER 11 AREAS OF PLANE FIGURES EXERCISE 45, Page Find the angles p and q in diagram (a) below. p = = 105 (interior opposite angles of a parallelogram are equal) q = = 35. Find the angles r and s in diagram (b) above. r = = 14 (the 38 angle is the alternate angle between parallel lines) s = = Find the angle t in diagram (c) above. t = =

2 EXERCISE 46, Page Name the types of quadrilateral shown in diagrams (i) to (iv) below, and determine for each (a) the area, and (b) the perimeter. (i) Rhombus (a) Area = = 14 cm (b) Perimeter = = 16 cm (ii) Parallelogram (a) Area = 30 6 = 180 (b) Perimeter = = 80 (iii) Rectangle (a) Area = = 3600 (b) Perimeter = ( 10) + ( 30) = 300 (iv) Trapezium = 190 (a) Area = cm (b) Perimeter = = = 6.91 cm 171

3 . A rectangular plate is 85 long and 4 wide. Find its area in square centimetres. 85 = 8.5 cm and 4 = 4. cm Area of plate = = 35.7cm 3. A rectangular field has an area of 1. hectares and a length of 150 m. If 1 hectare = m find (a) its width, and (b) the length of a diagonal. Area of field = 1. ha = m = 1000 m (a) Area = length width from which, width = (b) By Pythagoras, length of diagonal = area 1000 = 80 m length 150 = 170 m 4. Find the area of a triangle whose base is 8.5 cm and perpendicular height 6.4 cm. Area of triangle = 1 base perpendicular height = = 7. cm 5. A square has an area of 16 cm. Determine the length of a diagonal. A square ABCD is shown below of side x cm. The diagonal is given by length AC Area of square = x = 16 17

4 By Pythagoras, (AC) x x x = 16 from which, diagonal, AC = x [ 16] = 18 cm 6. A rectangular picture has an area of 0.96 m. If one of the sides has a length of 800, calculate, in millimetres, the length of the other side. Area = 0.96 m = and area = length breadth, i.e = 800 breadth from which, breadth = = Determine the area of each of the angle iron sections shown in below. (a) Area = (7 ) + (1 1) = = 9 cm (b) Area = (30 8) + 10(5 8 6) + (6 50) = = The diagram below shows a 4 m wide path around the outside of a 41 m by 37 m garden. Calculate the area of the path. 173

5 Area of garden = m Area of garden, neglecting the path = (41 8) (37 8) = 33 9 m Hence, area of path = (41 37) (33 9) = = 560 m 9. The area of a trapezium is 13.5 cm and the perpendicular distance between its parallel sides is 3 cm. If the length of one of the parallel sides is 5.6 cm, find the length of the other parallel side. Area of a trapezium = 1 (sum of parallel sides) (perpendicular distance between the parallel sides) i.e = 1 (5.6 + x) (3) where x is the unknown parallel side i.e. 7 = 3(5.6 + x) i.e. from which, 9 = x x = = 3.4 cm 10. Calculate the area of the steel plate shown below. Area of steel plate = (5 60) + (140 60)(5) (50 5) + = =

6 11. Determine the area of an equilateral triangle of side 10.0 cm. An equilateral triangle ABC is shown below. Perpendicular height, AD = by Pythagoras = cm Hence, area of triangle = 1 base perpendicular height = = cm 1. If paving slabs are produced in 50 by 50 squares, determine the number of slabs required to cover an area of m. Number of slabs = = 3 175

7 EXERCISE 47, Page A rectangular garden measures 40 m by 15 m. A 1 m flower border is made round the two shorter sides and one long side. A circular swiing pool of diameter 8 m is constructed in the middle of the garden. Find, correct to the nearest square metre, the area remaining. A sketch of a plan of the garden is shown below. Shaded area = (40 15) [(15 1) + (38 1) + (15 1) + 4 ] = 600 [ ] = = m = 48m, correct to the nearest square metre.. Determine the area of circles having (a) a radius of 4 cm (b) a diameter of 30 (c) a circumference of 00. (a) Area = r 4 = 50.7cm (b) Area = d d 30 r 4 4 = (c) Circumference = πr = 00, from which, radius, r = Hence, area = 100 r =

8 3. An annulus has an outside diameter of 60 and an inside diameter of 0. Determine its area. Area of annulus = d = 513 d If the area of a circle is 30, find (a) its diameter, and (b) its circumference. (a) Area of circle, A = d 4 i.e. 30 = d 4 from which, diameter, d = 430 = = 0.19 correct to decimal places. (b) Circumference of circle = r = d = = Calculate the areas of the following sectors of circles: (a) radius 9 cm, angle subtended at centre 75 (b) diameter 35, angle subtended at centre 4837' (a) Area of sector = r 9 = 53.01cm (b) Area of sector = r 60 = Determine the shaded area of the template shown below. 177

9 Area of template = shaded area = (10 90) = = An archway consists of a rectangular opening topped by a semi-circular arch as shown below. Determine the area of the opening if the width is 1 m and the greatest height is m. The semicircle has a diameter of 1 m, i.e. a radius of 0.5 m. Hence, the archway shown is made up of a rectangle of sides 1 m by 1.5 m and a semicircle of radius 0.5 m. Thus, area of opening = (1.5 1) = = 1.89 m 178

10 EXERCISE 48, Page Calculate the area of a regular octagon if each side is 0 and the width across the flats is The octagon is shown sketched below and is comprised of 8 triangles of base length 0 and perpendicular height 48.3/ Area of octagon = 8 0 = 193. Determine the area of a regular hexagon which has sides 5. The hexagon is shown sketched below and is comprised of 6 triangles of base length 5 and perpendicular height h as shown. Tan 30 = 1.5 h from which, h = 1.5 tan 30 = 1.65 Hence, area of hexagon = =

11 3. A plot of land is in the shape shown below. Determine (a) its area in hectares (1 ha = 10 4 m ), and (b) the length of fencing required, to the nearest metre, to completely enclose the plot of land. (a) Area of land = (30 10) = [ ] 9176 = 9176 m = ha = ha (b) Perimeter = = = 456 m, to the nearest metre

12 EXERCISE 49, Page The area of a park on a map is 500. If the scale of the map is 1 to determine the true area of the park in hectares (1 hectare = 10 4 m ) Area of park = m ha = 80 ha A model of a boiler is made having an overall height of 75 corresponding to an overall height of the actual boiler of 6 m. If the area of metal required for the model is 1500, determine, in square metres, the area of metal required for the actual boiler. The scale is : 1 i.e. 80 : 1 Area of metal required for actual boiler = m 80 = 80m 3. The scale of an Ordnance Survey map is 1:500. A circular sports field has a diameter of 8 cm on the map. Calculate its area in hectares, giving your answer correct to 3 significant figures. (1 hectare = 10 4 m ) Area of sports field on map = 8 d 10 m True area of sports field = m 4 ha = 3.14 ha 181