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KS3 Mathematics

KS3 Mathematics. S8 Perimeter, area and volume. S8 Perimeter, area and volume. Contents. S8.2 Area. S8.3 Surface area. S8.1 Perimeter. S8.4 Volume. S8.5 Circumference of a circle. S8.6 Area of a circle. Put these shapes in order. Perimeter. 1 cm.

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KS3 Mathematics

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  1. KS3 Mathematics S8 Perimeter, area and volume

  2. S8 Perimeter, area and volume Contents S8.2 Area S8.3 Surface area S8.1 Perimeter S8.4 Volume S8.5 Circumference of a circle S8.6 Area of a circle

  3. Put these shapes in order

  4. Perimeter 1 cm To find the perimeterof a shape we add together the length of all the sides. What is the perimeter of this shape? Starting point Perimeter = 3 + 3 + 2 + 1 + 1 + 2 3 = 12 cm 2 3 1 1 2

  5. Perimeter of a rectangle To calculate the perimeter of a rectangle we can use a formula. length, l width, w Using l for length and w for width, Perimeter of a rectangle = l + w + l + w = 2l + 2w or = 2(l + w)

  6. Perimeter 9 cm 5 cm 12 cm 4 cm Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given. For example, what is the perimeter of this shape? The lengths of two of the sides are not given so we have to work them out before we can find the perimeter. a cm Let’s call the lengths a and b. b cm

  7. Perimeter Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given. 9 cm a = 12 – 5 = 7 cm 5 cm 9 – 4 b = 12 cm 4 cm = 5 cm a cm 7 cm P = 9 + 5 + 4 + 7 + 5 + 12 = 42 cm b cm 5 cm

  8. Perimeter Calculate the lengths of the missing sides to find the perimeter. 5 cm p = 2 cm p 2 cm q = r = 1.5 cm q r s = 6 cm t = 2 cm s 6 cm u = 10 cm P = 5 + 2 + 1.5 + 6 + 4 + 2 + 10 + 2 + 4 + 6 + 1.5 + 2 4 cm 4 cm 2 cm 2 cm t u = 46 cm

  9. Perimeter What is the perimeter of this shape? Remember, the dashes indicate the sides that are the same length. 5 cm 4 cm P = 5 + 4 + 4 + 5 + 4 + 4 = 22 cm

  10. Perimeter What is the perimeter of this shape? Start by finding the lengths of all the sides. 4.5 cm 4.5 m Perimeter = 4.5 + 2 + 1 + 2 + 1 + 2 + 4.5 5 m 4 m = 17 cm 2 m 2 m 1 cm 1 cm 2 m

  11. Perimeter What is the perimeter of this shape? Before we can find the perimeter we must convert all the lengths to the same units. 256 cm In this example, we can either use metres or centimetres. 300 cm 3 m 1.9 m 190 cm Using centimetres, P = 256 + 190 + 240 + 300 2.4 m 240 cm = 986 cm

  12. Equal perimeters A C B C B A B C A Which shape has a different perimeter from the first shape? B A A

  13. S8 Perimeter, area and volume Contents S8.1 Perimeter S8.3 Surface area S8.2 Area S8.4 Volume S8.5 Circumference of a circle S8.6 Area of a circle

  14. Area Rug A Rug C Rug B The area of a shape is a measure of how much surface the shape takes up. For example, which of these rugs covers a larger surface?

  15. Area of a rectangle length, l Area of a rectangle = length × width width, w = lw 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 width. We can find the area of a rectangle by multiplying the length and the width of the rectangle together.

  16. Area of a rectangle What is the area of this rectangle? 4 cm 8 cm Area of a rectangle = lw = 8 cm × 4 cm = 32 cm2

  17. 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

  18. 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

  19. 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

  20. 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

  21. Area of shapes made from rectangles How can we find the area of the shaded shape? We can think of this shape as being made up of one rectangle cut out of another rectangle. 7 cm 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

  22. 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 = 11.5 units2

  23. Area of an irregular shapes on a pegboard A B D C How can we find the area of this irregular quadrilateral constructed on a pegboard? An alternative method would be to construct a rectangle that passes through each of the vertices. The area of this rectangle is 4 × 5 = 20 units2 The area of the irregular quadrilateral is found by subtracting the area of each of these triangles.

  24. 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 = 11.5 units2

  25. Area of an irregular shape on a pegboard

  26. Area of a triangle 1 2 What proportion of this rectangle has been shaded? 4 cm 8 cm Drawing a line here might help. What is the area of this triangle? Area of the triangle = × 8 × 4 = 4 × 4 = 16 cm2

  27. Area of a triangle

  28. 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,

  29. 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

  30. Area of a parallelogram

  31. 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,

  32. 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

  33. Area of a trapezium

  34. 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,

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. S8 Perimeter, area and volume Contents S8.1 Perimeter S8.2 Area S8.3 Surface area S8.4 Volume S8.5 Circumference of a circle S8.6 Area of a circle

  42. Surface area of a cuboid To find the surfacearea of a shape, 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.

  43. Surface area of a cuboid To find the surfacearea of a shape, 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.

  44. Surface area of a cuboid To find the surfacearea of a shape, 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.

  45. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. Can you work out the surface area of this cubiod? 5 cm 8 cm The area of the top = 8 × 5 = 40 cm2 7 cm The area of the front = 7 × 5 = 35 cm2 The area of the side = 7 × 8 = 56 cm2

  46. 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 7 cm + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2

  47. Formula for the surface area of a cuboid w l 2 × lw Top and bottom Front and back + 2 × hw 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 = = 2lw + 2hw + 2lh

  48. Surface area of a cube x Surface area of a cube = 6x2 How can we find the surface area of a cube of length x? All six faces of a cube have the same area. The area of each face is x × x = x2 Therefore,

  49. Chequered cuboid problem 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? 48 cm2

  50. Surface area of a prism What is the surface area of this L-shaped prism? 3 cm To find the surface area of this shape we need to add together the area of the two L-shapes and the area of the 6 rectangles that make up the surface of the shape. 3 cm 4 cm 6 cm Total surface area = 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15 5 cm = 110 cm2

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