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Measurement

Understand and convert different measurement units, calculate areas of shapes, work with composite shapes, and find total surface areas of 3D solids. Learn essential formulas with examples and practice exercises.

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Measurement

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  1. Measurement Chapter 6

  2. Converting units • When working in the topic of measurement, it is vital you have the right units • Units of measurement include: • Distance e.g. centimetres (cm), millimetres (mm), metres (m), kilometres (km) • Area (same as above but squared) e.g. cm2, mm2, m2, km2 • Volume (same as above but cubed) e.g. cm3, mm3, m3, km3 • At times you will need to switch between one unit to another e.g cm to m • Use the diagram below to help with this:

  3. Exercise 6A: Area • Area is the amount of flat space within a shape • Units are squared (2) • When you are figuring out the area of a shape: • Identify the shape • Write down the given measurements • Apply the appropriate formula • Remember to check units and how many decimal places are required

  4. Formulas to find area (p185-186) •  (Pi) is a number (use the shift button on your calculator to get it, and don’t round off until the end!) • l = length • w = width • r = radius (half diameter) • b = base • h = height • a/b/x etc. = different lengths within the shapes •  is the angle of the sector

  5. Heron’s formula • You can use this formula to find the area of a triangle if you are given the lengths of all three sides • a, b and c are the side lengths of the triangle • s is called the ‘semi-perimeter’ which means the perimeter of the triangle divided by two • Work out s first, then substitute s and the side lengths into the formula to find the area (A)

  6. Exercise 6A question 3(a) • Use Heron’s formula to find the area of the triangle: • First, write the measurements a = 5cm b = 12cm c = 16cm • Then, find ‘s’ s = (a+b+c)÷ 2 s = (5 + 12 + 16) ÷ 2 s = 33 ÷ 2 s = 16.5 • Then, apply the formula to find the area Area = Area = Area = Area = Area = 20.66cm2

  7. Composite shapes • These are unusual shapes, which are made through a combination of regular shapes • There are two ways to work these out: • Find the total area of all the shapes that make up the composite shape (e.g. the top example is made of two triangles added together) • Find the area of a larger shape, then subtracting the extra area involved (e.g. the bottom example is a large rectangle minus a smaller rectangle)

  8. Exercise 6A question 8(a) Find the area of the composite shape • First, what shapes are we dealing with here? • A rectangle (20cm by 15cm) • A circle (2 semi circles) with a diameter of 15cm (which means a radius of or 7.5cm) • The area of the circle needs to be subtracted from the area of the rectangle • Find the areas Rectangle area: • A = lw • A = 20 × 15 • A = 300cm2 Circle area: • A = r2 • A =  × 7.52 • A = 176.71cm2 • Find the total area of the composite shape Total area = rectangle – circle • A = 300 – 176.71 • A = 123.29cm2

  9. Exercise 6A question 9(a) Find the area of the composite shape • First, what shapes are we dealing with here? • A large circle (r = 7cm) • A small circle (r = 3cm) • The area of the small circle needs to be subtracted from the area of the large circle • Find the areas Large circle area: • A = r2 • A =  × 72 • A = 153.93804cm2 Small circle area: • A = r2 • A =  × 32 • A = 28.27433388cm2 Total area = large circle – small circle • A = 153.93804 – 28.27433388 • A = 125.66cm2

  10. Exercise 6A • Have a look through the rest of the questions – are there any others you would like to go through? • Otherwise – work on finishing the required questions for exercise 6A (p189) Questions 1, 3, 4, 5, 8, 9, 10, 13, 16 You don’t need to draw the shapes as long as you show your working clearly. You should draw the shape for worded problems (e.g. question 10) 6A should be finished by Wednesday, along with the conversion of units sheet

  11. 6B – total surface area (TSA) • The total surface area looks at 3D solid shapes (rather than just finding the area of a flat shape), finding the areas of each of the sides and adding these together • The units are squared (2) • Sometimes there is a formula to use, but sometimes the shapes will be trickier and you need to think about it or even sketch it to figure it out • A good tip for a complicated shape is to break it down into each of its sides • Work out the areas of the sides and then add them together

  12. Formulas for TSA • Rectangular prism (3D rectangular block) TSA = area of six sides added together TSA = 2(lh + lw + wh) • Cube TSA = area of six identical sides added together TSA = 6l2 • Sphere TSA = 4r2 • Cylinder TSA = 2 circles (top and bottom) + side TSA = 2r2 + 2rh • Cone TSA = circular base + curved pointed side TSA = r2 + rs (or) TSA = r(r + s) • Right pyramid (a 4 sided pyramid where the base is a square) TSA = area of base + area of four triangles (area of one triangle is ½bh) TSA = b2 + 2bh .

  13. Example: 6B question 1(a) and 2(a) • 1(a). Find the TSA • What shape is it? Cube • Formula for a cube? TSA = 6l2 TSA = 6(10)2 (put into calculator) TSA = 600cm2 • 2(a). Find the TSA • What shape is it? Sphere • Formula for TSA of a sphere? TSA = 4r2 TSA = 4(3)2 (put into calculator) TSA = 113.1cm2

  14. Example: 6B question 4(c) • Find the TSA • What shape do we have here? Not a ‘normal’ shape. We have two triangles (one on the top, one on the bottom) and 3 rectangles around the sides It is helpful to draw the shapes separately, find their areas then add them together • Triangles (x2) – base is 5.1cm, height is 8cm Area of one triangle = ½ bh Area = ½ x 5.1 x 8 Area = 20.4 (for one triangle) Area of both triangles = 20.4 x 2 = 40.8cm2 • Rectangles (x3) – 5.1x14, 7.2x14, 9.1x14 Area of rectangles = 71.4 + 100.8 + 127.4 Area of rectangles = 299.6cm2 • Total surface area = area of triangles + area of rectangles TSA = 40.8 + 299.6 TSA = 340.4cm2

  15. 6C – volume • Volume means the amount of space inside a solid shape • Measured in units cubed (e.g. cm3, m3)

  16. Volume of prisms • The volume of a prism is: Area of the base multiplied by the height Volume = bh (a prism is a shape where the top and bottom sides are the same)

  17. Volumes of other shapes

  18. Capacity • This just means the amount of liquid that could fit inside a solid shape • So rather than using cm3 or m3 etc., we use millilitres (mL), litres (L) or kilolitres (kL) • Use the following conversions to help converting to capacity: 1 cm3 = 1mL 1000 mL = 1L 1 m3 = 1000L 1000L = 1kL

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