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2. Area of Plane figures MATHEMATICS

3. Area of a rectangle length, l Area of a rectangle = length × breadth breadth, b = l x b Q.1 : Define area of rectangle We can find the area of a rectangle by multiplying the length and the breadth of the rectangle together. Area is measured in square units. For example, we can use mm2, cm2, m2 or km2. The 2 tells us that there are two dimensions, length and breadth.

4. rectangle l b The perimeter of a rectangle with length l and breadth b can be written as: Perimeter = 2l + 2b or Perimeter = 2 x (l + b) The area of a rectangle is given as: Area = l x b

5. Area of a rectangle 4 cm 8 cm Area of a rectangle = lw = 8 cm × 4 cm = 32 cm2

6. Area of shapes made from rectangles How can we find the area of this shape? We can think of this shape as being made up of two rectangles. 7 m Either like this … A 10 m … or like this. 15 m 8 m Label the rectangles A and B. B 5 m Area A = 10 × 7 = 70 m2 15 m Area B = 5 × 15 = 75 m2 Total area = 70 + 75 = 145 m2

7. Area of shapes made from rectangles 7 m A 10 m B 5 m 15 m Area A = 5 × 7 = 35 m2 Area B = 5 × 15 = 75 m2 Total area = 35 + 75 =110 m2

8. 6 cm 6 cm 2 cm 2 cm 4 cm 4 cm 12 cm 12 cm 8 cm 8 cm 4 cm 4 cm 2 cm 2 cm 8 cm2 12 cm2 16 cm2 24 cm2 2 cm 8 cm2 12 cm2 2 cm Area = 24 + 8 + 8 = 40 cm2 Area = 16 + 12 + 12 = 40 cm2

9. Area of shapes made from rectangles How can we find the area of the shaded shape? 7 cm We can think of this shape as being made up of one rectangle with another rectangle cut out of it. A 3 cm Label the rectangles A and B. 8 cm B 4 cm Area A = 7 × 8 = 56 cm2 Area B = 3 × 4 = 12 cm2 Total area = 56 – 12 = 44 cm2

10. Area of shapes made from rectangles A hotel is carpeting a function hall which has a wooden dance floor within it. Calculate the area of carpet required. 9cm This shape can be thought of as made up of one rectangle cut out of another rectangle. A 4cm Label the rectangles A and B. 11cm B Area A = 9 × 11 = 99cm2 6cm Area B = 4 × 6 = 24cm2 Total area = 99 – 24 = 75cm2

11. squares l When the length and the width of a rectangle are equal we call it a square. A square is just a special type of rectangle. The perimeter of a square with length l is given as: Perimeter = 4l The area of a square is given as: Area = l2

12. Area of a right-angled triangle Area of a triangle = × base × height 1 1 = bh 2 2 We can use a formula to find the area of a right-angled triangle: height, h base, b

13. Area of a right-angled triangle 1 2 What proportion of this rectangle has been shaded? 4 cm 8 cm What is the shape of the shaded part? What is the area of this right-angled triangle? Area of the triangle = × 8 × 4 = 4 × 4 = 16 cm2

14. Area of a right-angled triangle Area = × base × height 1 1 2 2 = × 8 × 6 Calculate the area of this right-angled triangle. To work out the area of this triangle we only need the length of the base and the height. 8 cm 6 cm 10 cm We can ignore the third length opposite the right angle. = 24 cm2

15. Area of a triangle Area of a triangle = × base × perpendicular height perpendicular height 1 1 base 2 2 Area of a triangle = bh The area of any triangle can be found using the formula: Or using letter symbols:

16. Area of a triangle b b h h h b Any side of the triangle can be taken as the base, as long as the height is perpendicular to it:

17. Area of a triangle Area of a triangle = bh = × 7 × 6 1 1 2 2 What is the area of this triangle? 6 cm 7 cm = 21 cm2

18. Area of an irregular shapes on a pegboard A D B E C How can we find the area of this irregular quadrilateral constructed on a pegboard? We can divide the shape into right-angled triangles and a square. Area A = ½ × 2 × 3 = 3 units2 Area B = ½ × 2 × 4 = 4 units2 Area C = ½ × 1 × 3 = 1.5 units2 Area D = ½ × 1 × 2 = 1 unit2 Area E = 1 unit2 Total shaded area = 10.5 units2

19. Area of an irregular shapes on a pegboard How can we find the area of this irregular quadrilateral constructed on a pegboard? Area A = ½ × 2 × 3 = 3 units2 Area B = ½ × 2 × 4 = 4 units2 A B Area C = ½ × 1 × 2 = 1 units2 Area D = ½ × 1 × 3 = 1.5 units2 Total shaded area = 9.5 units2 Area of irregular quadrilateral = (20 – 9.5) units2 C D = 10.5 units2

20. Area of a parallelogram Area of a parallelogram = base × perpendicular height perpendicular height base Area of a parallelogram = bh The area of any parallelogram can be found using the formula: Or using letter symbols:

21. Area of a parallelogram What is the area of this parallelogram? We can ignore this length 8 cm 7 cm 12 cm Area of a parallelogram = bh = 7 × 12 = 84 cm2

22. Area of a parallelogram What is the area of this parallelogram? We can ignore this length 8 cm 7 cm 12 cm Area of a parallelogram = bh = 7 × 12 = 84 cm2

23. Cubes and cuboids A cuboid is a 3-D shape with edges of different lengths. All of its faces are rectangular or square. How many faces does a cuboid have? Face 6 How many edges does a cuboid have? 12 How many vertices does a cuboid have? 8 Edge Vertex A cube is a special type of cuboid with edges of equal length. All of its faces are square.

24. Length around the edges or Length around the edges = 4(l + b + h) To find the length around the edges of a cuboid of length l, breadth b and height h we can use the formula: Length around the edges = 4l + 4b + 4h To find the length around the edges of a cube of length l we can use the formula: Length around the edges = 12l

25. Length around the edges 4 edges are 5 cm long. 3 cm 4 edges are 4 cm long. 4 cm 4 edges are 3 cm long. 5 cm Suppose we have a cuboid of length 5 cm, width 4 cm and height 3 cm. What is the total length around the edges? Imagine the cuboid as a hollow wire frame: The cuboid has 12 edges. Total length around the edges = 4 × 5 + 4 × 4 + 4 × 3 = 20 + 16 + 12 Perimeter = 48 cm

26. Surface area of a cuboid To find the surfacearea of a cuboid, we calculate the total area of all of the faces. A cuboid has 6 faces. The top and the bottom of the cuboid have the same area.

27. Surface area of a cuboid To find the surfacearea of a cuboid, we calculate the total area of all of the faces. A cuboid has 6 faces. The front and the back of the cuboid have the same area.

28. Surface area of a cuboid To find the surfacearea of a cuboid, we calculate the total area of all of the faces. A cuboid has 6 faces. The left hand side and the right hand side of the cuboid have the same area.

29. Formula for the surface area of a cuboid b l 2 × lb Top and bottom Front and back + 2 × hb h + 2 × lh Left and right side We can find the formula for the surface area of a cuboid as follows. Surface area of a cuboid = Surface area of a cuboid = 2lb + 2hb + 2lh

30. surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. So the total surface area = 5 cm 8 cm 2 × 40 cm2 Top and bottom 7cm + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2

31. surface area of a cuboid This cuboid is made from alternate purple and green centimetre cubes. What is its surface area? Surface area = 2 × 3 × 4 + 2 × 3 × 5 + 2 × 4 × 5 = 24 + 30 + 40 = 94 cm2 How much of the surface area is green? 47 cm2

32. Formula for the surface area of a cube Therefore, l Surface area of a cube = 6l2 How can we find the surface area of a cube of length l? All six faces of a cube have the same area. The area of each face is l × l = l2

33. surface area of a cube l = 4 cm Surface area of a cube = 6 x (l)2 = 6 x (4)2 = 6 x 4 x 4 = 96 cm3 Find the surface area of a cube of length 4 cm ?

34. Using nets to find surface area 6 cm 3 cm 3 cm 6 cm 5 cm 3 cm 3 cm It can be helpful to use the net of a 3-D shape to calculate its surface area. Here is the net of a 3 cm by 5 cm by 6 cm cuboid. Write down the area of each face. 18 cm2 Then add the areas together to find the surface area. 15 cm2 15 cm2 30 cm2 30 cm2 18 cm2 Surface Area = 126 cm2

35. Using nets to find surface area What is its surface area? Area of each face = ½bh = ½ × 6 × 5.2 = 15.6 cm2 5.2 cm Surface area = 4 × 15.6 = 62.4 cm2 6 cm

36. Area of a trapezium Area of a trapezium = (sum of parallel sides) × height a perpendicular height b 1 1 2 2 Area of a trapezium = (a + b)h The area of any trapezium can be found using the formula: Or using letter symbols:

37. Area of a trapezium = × 20 × 9 = (6 + 14) × 9 1 1 1 2 2 2 Area of a trapezium = (a + b)h What is the area of this trapezium? 6 m 9 m 14 m = 90 m2

38. Area of a trapezium = × 11 × 12 = (8 + 3) × 12 1 1 1 2 2 2 Area of a trapezium = (a + b)h What is the area of this trapezium? 8 m 3 m 12 m = 66 m2

39. Area problems This diagram shows a yellow square inside a blue square. What is the area of the yellow square? 3 cm 7 cm We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square. 10 cm If the height of each blue triangle is 7 cm, then the base is 3 cm. Area of each blue triangle = ½ × 7 × 3 = ½ × 21 = 10.5 cm2

40. Area problems 7 cm 10 cm This diagram shows a yellow square inside a blue square. What is the area of the yellow square? 3 cm We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square. There are four blue triangles so: Area of four triangles = 4 × 10.5 = 42 cm2 Area of blue square = 10 × 10 = 100 cm2 Area of yellow square = 100 – 42 = 58 cm2

41. Area formulae of 2-D shapes h b h b a h 1 1 2 2 Area of a triangle = bh Area of a trapezium = (a + b)h b You should know the following formulae: Area of a parallelogram = bh

42. Using units in formulae Remember, when using formulae we must make sure that all values are written in the same units. For example, find the area of this trapezium. 76 cm Let’s write all the lengths in cm. 518 mm = 51.8 cm 518 mm 1.24 m = 124 cm 1.24 m Area of the trapezium = ½(76 + 124) × 51.8 Don’t forget to put the units at the end. = ½ × 200 × 51.8 = 5180 cm2