= 10.2 km. (b) P = = 56 mm

Transcription

1 Perimeter Perimeter is the tota ength of the boundary of a shape. It is measured using km, m, cm or mm. We often use markings such as and to indicate equa engths when drawing diagrams or shapes. For exampe, two ines marked are the same ength. Worked Exampe Find the perimeter of each of the foowing shapes. (a) 1.4 km 2.8 km (b) 12 mm 1.7 km 1.1 km 16 mm Thinking 3.2 km (a) 1 Add the individua engths. (a) P = Write the tota ength, incuding the unit. = 10.2 km (b) 1 The sides marked with the same symbo have equa engths. Mutipy the engths by the number of times each appears. (b) P = Add the products. = Write the tota ength, incuding the unit. = 56 mm Perimeter formuas We can use formuas to find perimeters, using the same pronumera for sides of equa ength. Any pronumera can be used; however, we usuay use the etters (ength) and w (width) for rectanges, and b (base) for distances aong the bottom of some shapes, such as trianges or paraeograms. If there is no convenient formua to use, we simpy add a the side engths, using mutipication to simpify the cacuation when equa engths are added. Square Rhombus Rectange Paraeogram w a P = 4 P = 4 P = 2 + 2w or P = 2( + w) b P = 2a + 2b or P = 2(a + b) 5 Measurement 255

2 5.1 Worked Exampe 2 2 Use a formua to cacuate the perimeter of each of the foowing shapes. (a) 17 cm (b) 5 cm Thinking (a) 1 Write the formua for the perimeter of a rectange. 2 Substitute vaues for ength and width. 2.5 cm (a) P = 2 + 2w = Add the products. = Write the tota ength, incuding the unit. = 44 cm (b) 1 Let s be the ength of the side marked. Write a formua that mutipies s by the number of equa sides. (b) P = 5s 2 Substitute the side ength. = Write the product with units of ength. = 12.5 cm Answers page Navigator Perimeter Q1 Coumn 1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q11 Q1 Coumn 2, Q2, Q3, Q4, Q5, Q7, Q8, Q9, Q11 Q1 Coumn 3, Q2, Q3, Q5, Q7(b), Q8, Q9, Q10, Q12 1 Fuency 1 Find the perimeter of each of the foowing shapes. (a) 6.1 km (b) 11 m 4.5 km (c) 7 m 21 m 8.3 km 9.4 km 63 cm 70 cm (d) (e) (f) 5.25 m 6 cm 1.9 cm 256 PEARSON mathematics 8

3 5.1 (g) 25 mm (h) (i) 14 mm 55 mm 38 mm (j) 3 m (k) 60 cm () 25 cm 12 cm 30 cm 2 m 5 m 28 m 42 cm 25 m 30 cm 2 Use a formua to cacuate the perimeter of each of the foowing shapes. (a) (b) (c) cm 24 mm 1.2 cm (d) (e) (f) 3.3 cm Both of these are correct. The expression 2 + 2w is equivaent to 2( + w). 4.1 cm 45 m 25 mm 16 mm 28 m (g) (h) (i) 19 m 24 mm 43 m 4.9 km 68 mm 3 Find the perimeter of the foowing (draw a diagram if necessary). (a) a square of side ength 57 mm (b) a rectange of width 5.8 m and ength 10.7 m (c) a paraeogram of base 66 cm and a soping side ength of 32 cm (d) a rhombus of side ength 4.25 mm. 4 How far is it around the softba diamond shown? 5.6 km 2nd base 18.3 m 3rd base 1st base home base 5 Measurement 257

4 5.1 Understanding 5 Find the perimeter of each shape beow. A engths must have the same units before you can add them up. (a) (c) 2.8 cm 9 mm 80 cm (b) (d) 3.1 cm 4.1 cm 1.3 m 8 mm 1.3 m 70 cm (e) 800 m 2.1 km (f) 500 m 1.7 km 3.7 km 3.2 km 2.2 km 6 The roof pan for a new house appears here. Use it to cacuate the ength of guttering required if guttering is to be attached aong each edge of the roof. 5 m 19 m 1 m 3 m 9 m 27 m 7 (a) The ength of tape needed to go aong the perimeter of the figure shown in the diagram is: 60 cm A 190 cm 20 cm B 200 cm 20 cm C 220 cm 60 cm D 280 cm 30 cm (b) The perimeter of the tennis court shown here is: A m B m C m m D m 12.8 m m 258 PEARSON mathematics 8

5 5.1 Reasoning 8 A farmer needs to fence a new property. The fencing needs to go aong the boundary of the property, and aso divide it up into rectanges, as shown. 900 m 600 m (a) What is the tota ength of fence required? (b) If the type of fence used by the farmer costs $5.75 per metre, what wi it cost to fence the property? (c) If the fencing contractor can buid the fence at a rate of 20 metres per hour, how ong wi the job take? 9 (a) Find the side ength of a square if it has the same perimeter as a rectange of ength 19 cm and width 11 cm. (b) Find the width of a rectange if its ength is 28 cm and it has the same perimeter as a square of side 20 cm. 10 (a) Find the perimeter of a square tied area containing 25 square ties if each tie is 40 cm wide. (b) If the number of ties is doubed, find the perimeter of a the different rectanges that can be formed, using every tie. Open-ended 1000 m 11 Write possibe side engths for the triange, quadriatera and pentagon that the students are hoding beow. 12 Brett is designing a new vegetabe garden for his backyard. (a) Brett s friend Emma is heping him and needs to know the dimensions of the garden. Brett simpy tes Emma that the garden wi be in the shape of an isoscees triange, and wi have a perimeter of 56 m. Find three possibe sets of measurements for Brett s vegetabe garden if the side engths have to be whoe numbers of metres. (b) If Brett tes Emma that the perimeter can be any ength from 56 m to 100 m, find two new sets of dimensions for Brett s vegetabe garden if the two equa sides of the isoscees triange are to be twice as ong as the third side. A side engths need to be whoe metres. 5 Measurement 259

6 Circe reationships 5.2 circumference centre centre diameter centre radius Discovery Task The perimeter of a circe is caed the circumference. The diameter of a circe is any straight ine from the circumference through the centre to the circumference at the other side. The radius of a circe is any straight ine from the centre to the circumference. In order to cacuate the perimeters and areas of circes and shapes that have circuar parts, we need to understand how the circumference, diameter and radius of a circe are reated to each other. In this exercise, we wi investigate and estabish these reationships. 5.2 Navigator Circe reationships Q1, Q2, Q3, Q4, Q6(a), Q7, Q9 Q1, Q2, Q3, Q4, Q5, Q6(a), Q7, Q9 Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8 Q9 Answers page 617 Equipment required: Ruer, string, compass (optiona), circuar objects such as cups, pates, ids, drink bottes, whees (optiona) for Question 1; a scientific cacuator for Question 7(b) Fuency 1 Use your ruer to measure the diameter and radius of circes A F given on the foowing page. Record your measurements in the tabe underneath. Measure and record the circumference of each circe by carefuy aying the string aong it. (A cassmate can hep you do this reasonaby accuratey.) Mark the beginning and end of the circumference on the string, then hod it straight against a ruer and read off the ength of the circumference. Aternativey, choose six circuar objects, or use your compass to draw six circes of different sizes, then measure and record the diameter, radius and circumference using your ruer and string. (If you are using a compass, remember that the distance to which you open your compass is the same as the radius of the circe.) The diameter is the widest distance across the circe. 5 Measurement 261

7 5.2 A B C D E F Can t remember what unit ratio means? Turn to Section 4.3. Circe Radius r (cm) Diameter d (cm) Circumference C (cm) A B C D E F Understanding 2 (a) Use your competed tabe from Question 1 to describe any patterns or connections you can see between the numbers in the radius, diameter and circumference coumns. (b) To see the connections more ceary, add two coumns to the end of your tabe, with C the headings --- C and ---. Cacuate these two unit ratios for each circe that you d r measured, round off to two decima paces, and record your resuts in the new coumns. These unit ratios give us a scae factor for the ength of the circumference compared to the diameter or radius. 262 PEARSON mathematics 8

8 5.2 3 Use your competed tabe from Question 1 and the unit ratios you cacuated in Question 2 to copy and compete the foowing sentences. (a) The circumference of a circe is approximatey its diameter. (b) The circumference of a circe is approximatey (c) The ength of the diameter of a circe is Reasoning times the size of times the size of its radius. the ength of its radius. 4 Based on your answers to Questions 2 and 3, if a circe has a diameter of 4 m, predict the engths of the: (a) radius (b) circumference. 5 Based on your answers to Questions 2 and 3, predict what wi happen to: (a) the diameter when the radius of the circe is doubed. (b) the circumference 6 (a) Expain whether or not the measurements you obtained for the circumference by measuring with string and a ruer are accurate. C C (b) How woud your cacuations of --- and --- be affected if you measured the d r circumference as being: (i) smaer than actua size (ii) bigger than actua size? circumference 7 (a) The exact vaue of the ratio is represented by the symbo π (the Greek diameter etter pi). π is an irrationa number, which means that if written as a decima, it is a nonterminating, non-recurring decima. Describe what the digits in a non-terminating, non-recurring decima ook ike. (b) Find and press the π key on your cacuator. Write the vaue of π as a decima approximation, rounded to: Open-ended (i) two decima paces (ii) three decima paces (iii) four decima paces. 8 (a) Choose a ength for the diameter of a circe that ies in between the engths of two diameters isted in the tabe you drew in Question 1. (For exampe, if your tabe contains diameters of 4 cm and 6 cm, you coud choose 5 cm or 5.5 cm.) Use your circumference measurements for the two known diameters to estimate the circumference of a circe with your chosen diameter. (b) Repeat part (a) for a different vaue that ies in between two different diameters. 9 Cricket grounds vary in diameter and circumference. (For exampe, the SCG is much smaer than the MCG.) Estimate the diameter of a cricket ground (perhaps your schoo ova) and then find the approximate distance a cricket team wi run when they do a ap of honour after winning a championship on that ground. Assume the cricket ground is circuar. 5 Measurement 263