5

Measurement

255

Perimeter

Perimeter

is the total length of the boundary of a shape. It is measured using km, m, cm or mm. We often use markings such as and to indicate equal lengths when drawing diagrams or shapes. For example, two lines marked are the same length.

Perimeter formulas

We can use formulas to find perimeters, using the same pronumeral for sides of equal length. Any pronumeral can be used; however, we usually use the letters

l

(length) and

w

(width) for rectangles, and

b

(base) for distances along the bottom of some shapes, such as triangles or parallelograms. If there is no convenient formula to use, we simply add all the side lengths, using multiplication to simplify the calculation when equal lengths are added.

Worked Example 1

Find the perimeter of each of the following shapes.

(a) (b)

Thinking

(a) 1

Add the individual lengths.

(a)

P

=

1.7

+

1.4

+

2.8

+

1.1

+

3.2

2

Write the total length, includingthe unit.

=

10.2 km

(b) 1

The sides marked with the same symbol have equal lengths. Multiply the lengths by the number of times each appears.

(b)

P

=

2

×

12

+

2

×

16

2

Add the products.

=

24

+

32

3

Write the total length, including the unit.

=

56 mm

Square Rhombus Rectangle Parallelogram

P

=

4

lP

=

4

l P

=

2

l

+

2

w

or

P

=

2

(

l

+

w

)

P

=

2

a

+

2

b

or

P

=

2

(

a

+

b

)

1

1.4 km2.8 km

1.1 km

3.2 km

1.7 km

12 mm

16 mm

l

l

l

l l

l

w a

b

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256 PEARSON mathematics 8

Perimeter

Fluency1 Find the perimeter of each of the following shapes.

(a) (b) (c)

(d) (e) (f)

Worked Example 2Use a formula to calculate the perimeter of each of the following shapes.

(a) (b)

Thinking(a) 1 Write the formula for the perimeter

of a rectangle.(a) P = 2l + 2w

2 Substitute values for length and width.

= 2 × 17 + 2 × 5

3 Add the products. = 34 + 10

4 Write the total length, including the unit.

= 44 cm

(b) 1 Let s be the length of the side marked . Write a formula that multiplies s by the number of equal sides.

(b) P = 5s

2 Substitute the side length. = 5 × 2.5

3 Write the product with units of length.

= 12.5 cm

NavigatorQ1 Column 1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q11

Q1 Column 2, Q2, Q3, Q4, Q5, Q7, Q8, Q9, Q11

Q1 Column 3, Q2, Q3, Q5, Q7(b), Q8, Q9, Q10, Q12

2

17 cm

5 cm

2.5 cm

5.1Answerspage 617

16.1 km

8.3 km

4.5 km

9.4 km

7 m

21 m

11 m

63 cm

70 cm

5.25 m

6 cm1.9 cm

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5 Measurement

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257

(g) (h) (i)

(j) (k) (l)

2 Use a formula to calculate the perimeter of each of the following shapes.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

3 Find the perimeter of the following (draw a diagram if necessary).

(a) a square of side length 57 mm

(b) a rectangle of width 5.8 m and length 10.7 m

(c) a parallelogram of base 66 cm and a sloping side length of 32 cm

(d) a rhombus of side length 4.25 mm.

4 How far is it around the softball diamond shown?

25 mm

55 mm

38 mm

14 mm

28 m

25 m

2 m

5 m

3 m

25 cm

60 cm

30 cm

30 cm

12 cm

42 cm

2

9.5 cm

24 mm 1.2 cm

Both of these are correct.The expression 2l + 2w is equivalent to 2(l + w).

4.1 cm

3.3 cm

28 m

45 m25 mm

16 mm

43 m

19 m

5.6 km

4.9 km

68 mm

24 mm

2nd base

18.3 m

1st base3rd base

home base

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258 PEARSON mathematics 8

Understanding5 Find the perimeter of each shape below.

(a) (b)

(c) (d)

(e) (f)

6 The roof plan for a new house appears here. Use it to calculate the length of guttering required if guttering is to be attached along each edge of the roof.

7 (a) The length of tape needed to go along the perimeter of the figure shown in the diagram is:

A 190 cm

B 200 cm

C 220 cm

D 280 cm

(b) The perimeter of the tennis court shown here is:

A 23.77 m

B 29.255 m

C 58.51 m

D 69.48 m

All lengths must have the same units before you can add them up.

9 mm

2.8 cm

3.1 cm

8 mm

4.1 cm

80 cm

1.3 m70 cm

1.3 m

2.1 km

3.2 km3.7 km

800 m1.7 km

2.2 km

500 m

9 m

27 m

1 m19 m

3 m

5 m

60 cm

60 cm

20 cm20 cm

30 cm

12.8 m

5.485 m

10.97 m

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5 Measurement

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259

Reasoning8 A farmer needs to fence a new property. The fencing needs to go along the boundary

of the property, and also divide it up into rectangles, as shown.

(a) What is the total length of fence required?

(b) If the type of fence used by the farmer costs $5.75 per metre, what will it cost to fence the property?

(c) If the fencing contractor can build the fence at a rate of 20 metres per hour, how long will the job take?

9 (a) Find the side length of a square if it has the same perimeter as a rectangle of length 19 cm and width 11 cm.

(b) Find the width of a rectangle if its length is 28 cm and it has the same perimeter as a square of side 20 cm.

10 (a) Find the perimeter of a square tiled area containing 25 square tiles if each tile is 40 cm wide.

(b) If the number of tiles is doubled, find the perimeter of all the different rectangles that can be formed, using every tile.

Open-ended11 Write possible side lengths for the triangle, quadrilateral and pentagon that the students

are holding below.

12 Brett is designing a new vegetable garden for his backyard.

(a) Brett’s friend Emma is helping him and needs to know the dimensions of the garden. Brett simply tells Emma that the garden will be in the shape of an isosceles triangle, and will have a perimeter of 56 m. Find three possible sets of measurements for Brett’s vegetable garden if the side lengths have to be whole numbers of metres.

(b) If Brett tells Emma that the perimeter can be any length from 56 m to 100 m, find two new sets of dimensions for Brett’s vegetable garden if the two equal sides of the isosceles triangle are to be twice as long as the third side. All side lengths need to be whole metres.

900 m

1000 m

600 m

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5 Measurement 261

Circle relationships

In order to calculate the perimeters and areas of circles and shapes that have circular parts, we need to understand how the circumference, diameter and radius of a circle are related to each other. In this exercise, we will investigate and establish these relationships.

Circle relationships

Equipment required: Ruler, string, compass (optional), circular objects such as cups, plates, lids, drink bottles, wheels (optional) for Question 1; a scientific calculator for Question 7(b)

Fluency1 Use your ruler to measure the diameter and radius of circles A–F given on the

following page. Record your measurements in the table underneath. Measure and record the circumference of each circle by carefully laying the string along it. (A classmate can help you do this reasonably accurately.) Mark the beginning and end of the circumference on the string, then hold it straight against a ruler and read off the length of the circumference.

Alternatively, choose six circular objects, or use your compass to draw six circles of different sizes, then measure and record the diameter, radius and circumference using your ruler 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 circle.)

The perimeter of a circle is called the circumference.

The diameter of a circle is any straight line from the circumference through the centre to the circumference at the other side.

The radius of a circle is any straight line from the centre to the circumference.

NavigatorQ1, Q2, Q3, Q4, Q6(a), Q7, Q9 Q1, Q2, Q3, Q4, Q5, Q6(a), Q7,

Q9Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8 Q9

Discovery Task

circumference

centrediametercentre radiuscentre

5.2Answerspage 617

The diameter is the widestdistance across the circle.

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262 PEARSON mathematics 8

Understanding2 (a) Use your completed table from Question 1 to describe any patterns or connections you

can see between the numbers in the ‘radius’, ‘diameter’ and ‘circumference’ columns.

(b) To see the connections more clearly, add two columns to the end of your table, with

the headings and Calculate these two unit ratios for each circle that you

measured, round off to two decimal places, and record your results in the new columns. These unit ratios give us a scale factor for the length of the circumference compared to the diameter or radius.

Circle Radius r (cm) Diameter d (cm) Circumference C (cm)

A

B

C

D

E

F

A B C D E F

Can’t remember what ‘unit ratio’ means? Turn to Section 4.3.

Cd--- C

r--- .

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5 Measurement

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263

3 Use your completed table from Question 1 and the unit ratios you calculated in Question 2 to copy and complete the following sentences.

(a) The circumference of a circle is approximately times the size of its diameter.

(b) The circumference of a circle is approximately times the size of its radius.

(c) The length of the diameter of a circle is the length of its radius.

Reasoning4 Based on your answers to Questions 2 and 3, if a circle has a diameter of 4 m, predict the

lengths of the:

(a) radius (b) circumference.

5 Based on your answers to Questions 2 and 3, predict what will happen to:

(a) the diameter (b) the circumference

when the radius of the circle is doubled.

6 (a) Explain whether or not the measurements you obtained for the circumference by measuring with string and a ruler are accurate.

(b) How would your calculations of and be affected if you measured the circumference as being:

(i) smaller than actual size (ii) bigger than actual size?

7 (a) The exact value of the ratio is represented by the symbol π (the Greek letter pi).

π is an irrational number, which means that if written as a decimal, it is a non-terminating, non-recurring decimal. Describe what the digits in a ‘non-terminating, non-recurring’ decimal look like.

(b) Find and press the ‘π’ key on your calculator. Write the value of π as a decimal approximation, rounded to:

(i) two decimal places (ii) three decimal places (iii) four decimal places.

Open-ended8 (a) Choose a length for the diameter of a circle that lies in between the lengths of two

diameters listed in the table you drew in Question 1. (For example, if your table contains diameters of 4 cm and 6 cm, you could choose 5 cm or 5.5 cm.) Use your circumference measurements for the two known diameters to estimate the circumference of a circle with your chosen diameter.

(b) Repeat part (a) for a different value that lies in between two different diameters.

9 Cricket grounds vary in diameter and circumference. (For example, the SCG is much smaller than the MCG.) Estimate the diameter of a cricket ground (perhaps your school oval) and then find the approximate distance a cricket team will run when they do a lap of honour after winning a championship on that ground. Assume the cricket ground is circular.

Cd--- C

r---

circumferencediameter

-----------------------------------

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