Lesson 21. Chapter 2: Perimeter, Area & Volume. Lengths and Areas of Rectangles, Triangles and Composite Shapes

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1 ourse: HV Lesson hapter : Perimeter, rea & Volume Lengths and reas of Rectangles, Triangles and omposite Shapes The perimeter of a shape is the total length of its boundary. You can find the perimeter of a shape by adding the lengths of its sides. Length l Width w w Perimeter of rectangle = l + l + w + w = (l + w) l The area of a shape is a measure of the amount of space it covers. Typical units of area are square centimetres (cm ), square metres (m ) and square kilometres (km ). l w rea of rectangle = l w You can find the area of a rectangle using the following formula: rea of rectangle = length width = l w Perimeter, rea & Volume Page

2 ourse: HV rea of a parallelogram You can use the rectangle formula to find the area of a parallelogram: ny side of a parallelogram can be called its base. The perpendicular distance between the base and its opposite side is called the perpendicular height (or height) of the parallelogram. cm E 3cm 5cm The base of parallelogram is 5cm long and the perpendicular height is. E is cm. 5cm E 3cm cm You can cut off the triangle E from one end of the parallelogram and place it at the other end. is equal and parallel to so they fit exactly together. E 5cm The new shape is the rectangle. The area of the original parallelogram is the same as the area of this rectangle, 5, which is the base multiplied by the height. rea of parallelogram = base height = b h Perimeter, rea & Volume Page

3 ourse: HV rea of a triangle There are two ways of finding how to calculate the area of a triangle. First, if we think of a triangle as half a parallelogram we get rea of triangle = area of parallelogram = (base height) Second, if we enclose the triangle in a rectangle we see again that the area of the triangle is half the area of the rectangle. rea of triangle = area of rectangle = (base height) In either case, the equation to find the area of the triangle is the same. rea of triangle = (base height) Perimeter, rea & Volume Page 3

4 ourse: HV rea of a Trapezium It is useful to know how to find the area of a trapezium with a simple formula instead of finding the areas of the two triangles and then add. So, let s construct an equation to find the area of the trapezium. onsider this shape q h p This is a Trapezium. Note that is parallel to but their lengths are not equal. = q and = p We know how to work out the area of triangle and triangle. rea of triangle = (base height) = rea of triangle = (base height) = p h q h The heights of both triangles are the same, as each is the distance between the parallel sides of the trapezium. Therefore total area of = ph qh ( p q) h the area of a trapezium is equal to (sum of parallel sides) (distance between them) Perimeter, rea & Volume Page 4

5 ourse: HV Example E is a square of side 8 cm. The total height of the shape is cm. Find the area of E. cm E 8 cm nswer 8 cm In order to find the area of a compound shape, first we have to split it in shapes that we know how to find the area of. This compound shape can be divided into a square E and a triangle E. So rea of triangle E = (base height) 8 4 cm = 6 cm rea of Square E = 8 8 cm = 64 cm So Total rea = rea of triangle E + rea of Square E = 6 cm + 64 cm = 80 cm Perimeter, rea & Volume Page 5

6 ourse: HV Example Work out the area of this shape. nswer First method ivide the shape into parts whose areas you can find. This can be done in several ways. One way is: rea of rectangle = l w = 7 4 = 8cm a b h cm rea of trapezium = So the shaded area is = 46cm 7cm 7cm 8cm 8cm cm cm Second Method dd on a small triangle to make a larger rectangle. 7cm rea of large rectangle = l w = 7 8 = 56cm rea of small triangle = base height = 5 4 0cm So the shaded area is 56 0 = 46cm 8cm cm 5cm Example 3 The area of a triangle is 0 cm. The height is 8 cm. Find the length of the base. nswer Let the base be b cm long. rea of triangle = (base height) 8 cm So, the base is 5 cm long. 0 b = 4b b = 5 b cm Perimeter, rea & Volume Page 6

7 ourse: HV Exercise. Find the area of each of the following trapeziums: a) b) 4 cm 8.5 cm 6 cm 3 cm 0 cm c) d) 5.5 cm 4 cm 9 cm 7 cm 8 cm.5 cm 3 cm. Find the areas of the following triangles. If necessary, turn the page round and look at the triangle from a different direction. a) b) 4 cm cm 5 cm 7 cm 7 cm 6 cm Perimeter, rea & Volume Page 7

8 ourse: HV 3. Find the areas of the following parallelograms: a) b) 40 cm 0 cm cm 9 cm 0 cm 4. Find the area of the following figures in square centimetres. The measurements are all in centimetres. a) b) Find the missing measurements of the following shapes. rea ase Height a. Triangle 4 cm 6 cm b. Parallelogram 36 cm 70 cm c. Rectangle.8 m 0.64 m 6. Find the area of each of the compound shapes. a) b) 9 cm 6 cm 3cm 4 cm 3 cm cm 4 cm Perimeter, rea & Volume Page 8

9 ourse: HV c) d) is a rhombus = 9 cm. = cm. is a kite ( is the axis of symmetry. The diagonals cut at right angles.) = 0 cm. = cm. 7. For these shapes calculate (i) the perimeter (ii) the area. (a) (b) (c) 8 cm 6.5 cm 8.5 cm 6 cm 0 cm 4 cm 6.5 cm 3.5 cm 4 cm 3 cm 8. Work out the areas of these shapes correct to d.p. (a) 5.4 m (b) 4.9 m 5.5 cm 6.5 cm Perimeter, rea & Volume Page 9

10 ourse: HV 9. alculate the area of these shapes correct to d.p. (a) (b) (c).4 cm.7 cm 7.5 cm 6.78 cm 7.6 cm 4. cm 4.5 cm 0. The diagram shows the measurements, in inches, of the L on an L plate. Work out the area of the L..5 in 4 in 3.5 in.5 in. Work out the area of the shape EF. F 7 cm E 30 cm 9.5 cm cm. landscape gardener designs a layout for a garden with an awkward shape. The diagram shows his plan. Find the area of the lawn. m 5 m 3 m Patio.6 m 4 m 3.4 m Shrubs Lawn 3 m Flowers.6 m 3.6 m 4.6 m 3.6 m Perimeter, rea & Volume Page 0

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