Measurement Note 1 : Measurement Systems In NZ the measurement system used is the metric system. The units relate directly to each other. CapacityBasic UnitSymbol Distance Mass (weight) Capacity Temperature Time Area Land Area Volume metrem gramg L C s/min degrees celsius litre seconds/minutes square metres m hectares ha cubic metres m To change within a unit from one prefix to another prefix, we either multiply or divide by a power of 10. smaller to larger unit divide by a power of 10 larger to smaller unitmultiply by a power of 10 Examples: convert the following 5.76m to cm 489mL to L cm to km Homework Book Page STARTERS Convert the following: 59mL to L 4200kg to tonne 11m465mm to cm A dairy stores milk in 5 litre containers. How many 350mL milkshakes can be made from one of these containers? Note 2: Derived Units Derived units show comparisons between two related measures. For example, speed is a measure of how much distance changes over time. The units for speed are m/s or km/h. Distance SpeedTime Examples: A cyclist travels at a steady speed of 24km/h for 40 minutes. How far did the cyclist travel? 40 minutes = 2/3hour Distance = speed x time = 24 x 2/3 = 16 km Changing from one speed unit to another Note: 1km = 1000m 1 hour = 3600sec Examples: Change 45km/h into m/s 45km/h= 45 x 1000m/h = 45000m/h = 45000m/3600s = 12.5 m/s Examples: Change 74m/s into km/h 74m/s= 3600x74m/h = m/h = 266.4km/h Homework Book Page STARTERS Convert the following: 19m/s to km/h A truck travels at an average speed of 75km/h for a distance of 300km. What time does the journey take? A Boeing 747 has a cruising speed of 910km/h. Change this into m/s? Note 3: Perimeter The perimeter is the distance around the outside of a shape. Start at one corner and work around the shape calculating any missing sides. 6 cm 5 cm 2 cm 5 cm Perimeter = 5cm + 3cm + 6cm + 2cm + 11cm + 5cm = 32cm Homework Book Page STARTERS Calculate the perimeter of The plan shows an L-shaped paddock. Calculate the total cost of fencing it at $24/m Note 4: Circumference The perimeter of a circle is called the circumference. The formula for the circumference is: C = d or C = 2r where d = diameter r = radius. Example: Find the circumference of C = 2r = 2 x x 8cm = 50.3cm (1dp) If a sector has an angle at the centre equal to x, then the arc length is x / 360 of the circumference. Example: Find the perimeter of the sector Angle of sector = 360 = 240 Arc Length = x / 360 x 2r = 240 / 360 x 2 x x 6m = 25.1m (1dp) Perimeter = 2 x 6m m = 37.1m Homework Book Page STARTERS Calculate the perimeter of Paul goes for a short cycle ride. Each wheel on his bike has a radius of 27cm. His distance counter tells him the wheel has rotated 650 times. Find how far he has travelled in metres. Note 5: Area Area is measured in square units. Examples converting units: Examples of converting units 5.6cm 2 to mm 2 Big Small x 5.6cm 2 = 5.6 x 100 = 560mm cm 2 to m 2 Small Big cm 2 = 10000 = 39.6m 2 Examples: Calculate the area of these shapes 7m7m 9m9m 17m 12m 8m8m 15m 17m 10m 12 7 = 42 m 2 Area = base height = 17 10 = 85 m 2 Radius = 7 2 = 3.5 cm Area = x / 360 x x r = 180 / 360 x x 3.5 = 19.2 m (1dp) Area = (sum of bases) x height = (9 + 12) x 7 = 73.5 m (1dp) Homework Book Page 170 171 STARTERS Find the area of A chocolate bar is wrapped in a rectangular piece of foil measuring 10cm by 15cm. Calculate the area of the piece of foil. How many pieces could be cut out from a larger sheet of foil measuring 120cm by 75cm? Note 6: Compound Area Compound shapes are made up of more than one mathematical shape. To find the area of a compound shape, find the areas of each individual shapes and either add or subtract as you need to. Examples: find the area of Area splits into a rectangle and a triangle Area = Area rectangle + area triangle = b h + b h = 4 5 + 4 2 = 24cm 2 Area splits into a rectangle with another rectangle taken away Area = area big rectangle area small rectangle = b h - b h = 6 4 3 2 = 18m 2 Homework Book Page 172 174 STARTERS Find the area of Trapezium = 750 Rectangle = 1000 Half Circle = Area = = cm 2 Note 7: Finding missing parts of shapes To find missing sides of shapes, rearrange the formulas. Example 1: The area of the triangle is 135m 2. Calculate the height of the triangle. Area = base height 135= 18 x 135= 9x x = 15m Example 2: Calculate the radius of a circle with an area of 65cm 2. Area = r 2 65= r 2 r 2 = 65 / r = 65 / = 4.5 cm EXERCISES: Each of these shapes has an area of 60cm 2. Calculate the lengths marked x. 15cm 10cm 60 =7.7cm 2.5cm EXERCISES: Calculate the radii of these circles with the given areas m 18.7 cm 1.38 cm0.798 km EXERCISES: A circle has an area of 39.47m 2. Calculate: Radius Diameter circumference 3.55 m 7.09 m m