Ch06

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Chapter 6 Perimeter and Area This chapter deals with the use of formulas to find the area of quadrilaterals, and the perimeter and area of composite figures. At the end of this chapter you should be able to: ✓ develop and use formulas to find the area of quadrilaterals ✓ calculate the area and perimeter of composite figures including quadrants and semicircles ✓ calculate the perimeter and area of sectors and composite figures involving sectors.

Syllabus reference MS5.1.1, 5.2.1 WM: S5.2.1–S5.2.5

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Diagnostic Test

1

The formula A =

1 --2

h(a + b) could be used

to find the area of a: A kite

B trapezium

C rhombus

D all of these

5

A 30.38 cm2

B 15.19 cm2

C 11.1 cm2

D 22.2 cm2

The area of this quadrilateral is closest to: B

2

5.3 cm

9.8 cm

3

6 cm 20 cm

>

D

A AC = 20 cm

2.2 cm

2

C

4 cm

The area of this trapezium is:

> 6

A 16.61 cm

B 33.22 cm2

C 57.134 cm2

D 114.268 cm2

A 480 cm2

B 200 cm2

C 300 cm2

D 100 cm2

The perimeter of this semicircular garden is:

The area of this kite is: 5.2 m

3.8 cm

7

A 9.3 m

B 21.5 m

C 13.4 m

D 4.08 m

The perimeter of this shape is:

8.5 cm

B 24.6 cm

2

D 16.15 cm2

A 32.3 cm C 12.3 cm 4

7.1 m

2

2

1.8 m

The area of this rhombus is closest to:

8

4.9 cm 6.2 cm

A 19.2 m

B 20.6 m

C 23.5 m

D 12.78 m

Simone has 50 m of garden edging. The radius of the circular garden she can enclose with the edging is: A 15.9 m

B 7.96 m

C 12.5 m

D 157 m

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

9

10

The perimeter of a rhombus with diagonals 4.1 cm and 6.3 cm is closest to:

13

The perimeter of this sector is closest to: A 7.7 cm

A 3.76 cm

B 7.5 cm

B 23.7 cm

C 15 cm

D 30 cm

C 440 cm

55°

D 25.8 cm

The area of this figure is closest to:

8 cm 8.2 cm

14 5.8 cm

The area of this sector is closest to: A 212 cm2 B 248 cm2

3.4 cm

C 59.6 cm2

11

A 161.7 cm2

B 59.12 cm2

C 36 cm2

D 34.8 cm2

75°

D 1350 cm2

18 cm

The shaded area is closest to:

10.1 m

Use this diagram to answer questions 15 and 16.

4.8 cm

3m 2

A 248.1 m

B 62 m

2

D 46.8 m2

C 88.2 m 12

25o

2

1.2 m

A farmer fertilises a paddock consisting of a rectangle and a semicircle, shown below. The fertiliser is spread at the rate of 2.3 kilograms per square metre. The amount of fertiliser the farmer needs is closest to:

15 m

15

16 28 m

A 2.6 t

B 1.2 t

C 250 kg

D 966 kg

The perimeter of the shape is closest to: A 29.2 m

B 9.1 m

C 12.7 m

D 9.7 m

The area of the shape is closest to: A 5.56 m2

B 4.25 m2

C 1.96 m2

D 10.36 m2

If you have any difficulty with these questions, refer to the examples and questions in the sections listed in the table. Question Section

1–5

6–9

10, 11

12

13–16

B

C

D

E

F

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A. AREA AND PERIMETER REVIEW This section reviews conversions, area and perimeter from Stage 4.

Linear conversions 1 km = 1000 m

1 m = 100 cm

1 cm = 10 mm

Example 1 Convert: a 50 mm to cm

b 8.6 km to m

a

b

50 mm 50 = ------ cm 10 = 5 cm

8.6 km 8.6 × 1000 m = 8600 m

Area conversions When converting area units, which are square units, the linear conversion must be squared. Since 10 mm = 1 cm then 102 mm2 = 12 cm2

(squaring both sides)

∴ 100 mm2 = 1 cm2 The hectare (ha) is a special unit of area: 1 ha = 10 000 m2.

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Example 2 Convert: a 25 cm2 to mm2 a

b 2000 cm2 to m2

1 cm = 10 mm ∴ 1 cm2 = 102 mm2 ∴ 1 cm2 = 100 mm2 ∴ 25 cm2 = 2500 mm2 2

b

100 cm = 1 m 1002 cm2 = 12 m2 ∴ 10 000 cm2 = 1 m2 ∴ 2000 cm2 = 0.2 m2 (dividing both sides by 5)

Exercise 6A 1

2

Convert: a 21 cm to m d 4 cm to mm g 200 mm to cm j 8.3 cm to mm m 0.05 km to cm

b e h k n

180 mm to cm 2.3 m to cm 280 cm to m 6.3 km to m 3.2 m to mm

c f i l o

3500 m to km 1.8 km to m 5.2 m to cm 0.03 m to cm 83 000 cm to km

Convert: a 4 cm2 to mm2 d 32 km2 to m2 g 5 ha to m2

b 31 m2 to cm2 e 40 000 cm2 to m2 h 7.3 ha to m2

c f i

5.3 m2 to mm2 7 000 000 mm2 to m2 42 000 m2 to ha

When converting square units, square the conversion first.

3

Convert: a 15 cm2 to mm2 c 32 000 cm2 to m2 e 235 m2 to cm2 g 7.82 m2 to ha i 23 km2 to ha k 5.2 m2 to cm2

b d f h j l

15 ha to m2 3280 mm2 to cm2 36.5 ha to m2 3 654 200 cm2 to ha 0.004 2 ha to cm2 0.002 m2 to mm2

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Example 3

The perimeter of a circle is called the circumference.

Find the circumference of this circle to 1 decimal place. C = 2πr C = 2 × π × 8.2 = 51.5221... = 51.5 cm

4

8.2 cm

Find the circumference of these circles, to 1 decimal place if necessary. a

Radius is half the diameter.

b

0.4 km

5.1 cm

c

d 12.6 cm 48 cm

Example 4 Find the area of this rectangle. A = lb = 3.4 × 6.8 cm2 = 23.12 cm2

3.4 cm

6.8 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

5

Find the area of each rectangle. a

b

c

d

12 m 16 cm 8 cm

12 m

15 m 4 cm 5 cm

Example 5 Find the area of each triangle. a

b 3m

8m

5m

2.5 cm

c

6 cm 12 m

a A = 1--2- bh

b A = 1--2- bh

c A = 1--2- bh

=

=

=

1 --2

× 12 × 5 m2

= 30 m2

6

× 8 × 3 m2

1 --2

= 12 m2

1 --2

× 2.5 × 6 cm2

= 7.5 cm2

Find the area of each triangle. a

b

c

d

7m 4 cm

4m

40 m

8 cm

2 cm 9 cm

32 m

Example 6 Find the area of the parallelogram. A = bh = 10 × 5 cm2 = 50 cm2

5 cm

10 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

7

Find the area of each parallelogram. a b

c 6 cm

3 cm

10 cm 6 cm 12 cm

8 cm

Example 7 Find the area of these circles correct to 1 decimal place. a

b 12 cm

5 cm

a Area = πr2 = π × 5 × 5 cm2 = 25π cm2
78.5 cm2

8

b Area = πr2 = π × 6 × 6 m2 = 36π m2
113.1 m2

Find the area of each circle correct to 2 decimal places. a b c 8 cm 14 m

d

e

2 6.

cm

8.5 cm

f

15.3 m

1.26

km

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Investigation 1 WM: Applying Strategies

Formulas for area In Stage 4 you learnt that the area of a triangle is: A = 1--2- b × h Use this formula to find an expression for the area of a rhombus and a kite. 1

Rhombus Use this diagram to find an expression for the area of a rhombus with diagonals x and y units in length. y y 2

x

y 2

x

2

Kite The formula for the area of a kite is the same as that of a rhombus. Compare this derivation with yours from question 1.

1 2

Divide into two triangles. y cm

y cm

x cm

Area =

1 --2 1 --4 1 --2

×

( 1--2-

y) × x +

1 --2

×

( 1--2-

y) × x

1 2

y cm

xy + xy B. AREAx cmOF SPECIAL =QUADRILATERALS 1 --4

= xy

From Investigation 1, the following formulas have been developed. Use the formula for the area of a triangle to find an expression for the area of a trapezium.

Rhombus Trapezium 3 Use these diagrams to find an expression for the area of a trapezium. a

a

t

cu

h

b

b

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B. AREA OF SPECIAL QUADRILATERALS From Investigation 1, we have developed the following formulas.

Rhombus y

x

Area =

1 -2 1 -2

× product of the lengths of the diagonals

A = xy, where x and y are the lengths of the diagonals

Kite y cm

Area

A=

1 -2

xy

x cm

Trapezium a

A = 1--2- ah + 1--2- bh ∴ A = 1--2- h (a + b)

height

a+b or A =  ------------ h  2  b

Note the ‘height’ is the perpendicular distance between the two parallel sides. Sometimes it is a side but usually it is not.

Example 1 Find the area of the rhombus with diagonals of length 5 cm and 7 cm. Area = 1--2- xy =

1 --2

× 5 × 7 cm2

= 17.5 cm2

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Exercise 6B 1

Find the area of each rhombus. a b

c 4.3 cm 6.3 m

4 mm 12 mm

7.2 cm

9.5 m

Example 2 Find the area of this kite. Area = 1--2- xy =

1 --2

5 cm

×5×8

8 cm

= 20 cm2

Find the area of each kite. a

b

4.8 m

6 cm

c

15 cm

9 cm

2

11.6 m

4.3 cm

Example 3 Find the area of this trapezium. First identify the height then use the formula. Area = 1--2- h(a + b) =

1 --2

× 4(11 + 16)

= 54 m2

11 m 4m 16 m

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

3

Find the area of each trapezium. Identify the height first. a

The height is perpendicular to the parallel sides.

b

6m 4m

6 cm

3 cm

7 cm

10 m

c

d

16 cm 7 cm

35 mm

50 mm

12 m 28 mm

Example 4 Find the area of this quadrilateral.

Q 3 cm

Area = Area of triangle A + Area of triangle B =

1 --2

× 14 × 3 +

1 --2

A

× 14 × 5

5 cm 14 cm B

= 56 cm2 P

4

Find these areas. a

b

4 cm P

c

4 cm PQ = 8 cm

RS = 13 m

11 m

Use the correct formula to find the area of these quadrilaterals. a b c 10 m

8 cm

8 mm

12 cm

5

8m

6m

15 mm 7m

TU = 28 m

S

Q 1 cm

T

5 cm

R

U

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

d

e

f

3 km 11.3 m

18 m 15 km

7.5 m

8 km 8.5 m 4.1 m

C. PERIMETER OF COMPOSITE FIGURES This section involves finding the perimeter of composite figures, and the solution of worded problems involving perimeter. Composite figures are those made up of more than one plane shape, including curved shapes.

Example 1 Find the perimeter of these figures. a

b 6 cm

40 m

a Perimeter Divide by 2 because it is a = (2πr ÷ 2) + 6 semicircle. =π×3+6 = 3π + 6
15.4 cm (1 d.p.) b Perimeter = circumference of circle + 2(length of straight side) = π × 40 + 2 × 40 = 40π + 80
205.7 m Two semicircles make a circle.

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Exercise 6C 1

Find the perimeter of these, giving answers to one decimal place. a b 10 cm 8 cm

c

d

20 m

20 m

20 m

18 m

Example 2

A quarter of a circle, so divide by 4.

Find the perimeter of these figures. a

b

8 cm

a

2

Perimeter = (2πr ÷ 4) + 2 × 8 = (2π × 8 ÷ 4) + 2 × 8 = 28.6 (1 d.p.)

12 m

b

Perimeter = (2πr ÷ 4) + 4 × 12 = (2 × π × 12 ÷ 4) + 4 × 12 = 66.8 (1 d.p.)

Find the perimeter of these, correct to one decimal place. a b c

d 20 cm

18 mm

10 cm

6 cm 10 m

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3

Find the perimeter of the following (giving your answer correct to 4 significant figures). a b c

4 cm 6 cm 4m

4

5

a i Find the lengths of the paths from A to B along the 4 small semicircles, and along the larger semicircle (to 2 d.p.). ii Which is shorter? b What is the difference between the lengths of the two paths?

A

B 8m

Find the perimeter correct to 2 decimal places (all measurements are in cm). a b c 10

8

11

15 5

7

A farmer decides to fence a 400 m by 350 m paddock with a 4-strand wire fence. Find the total cost of the wire required given that single strand wire costs 12.4 cents per metre.

6

7

Find the total length of string used to tie a box as illustrated. An extra 15 cm is required for the knot and bow.

10 cm

20 cm 15 cm

8 2m

The framing of a toolshed consists of square galvanised tubing which costs $4.65 a metre. Find the total cost of the tubing necessary to make the framing of the shed opposite.

4m 3m

9

A garden consists of six rectangular-shaped 8 m by 7 m garden beds and a 2 m wide path surrounding them as shown. Jarrah timber strips are used to surround each bed and the whole garden area. Find the total length of jarrah required.

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Example 3 Izat ran around a circular track. He ran 500 m. Find the radius of the track. C = 2πr r

500 2×π×r ------------ = --------------------2×π 2×π 500 ∴ r = ----------------(2 × π) = 79.6 cm (to 1 d.p.)

10

A circular plate has circumference 50 cm. Find the radius correct to 1 d.p.

11

Georgette wants a circular track with a circumference of 200 m. Find the radius of the track correct to 1 d.p.

12

A satellite has a circular orbit 800 km above the Earth’s surface. a If the radius of the Earth is 6400 km, find the radius of the orbit of the satellite. b Find the circumference of the satellite’s orbit. c If the satellite makes one orbit in a day, find the speed of the satellite.

13

A bicycle wheel has diameter of 0.6 m. Through how many complete revolutions must the wheel turn during a 100 km trip?

14

A newspaper company decides to place a plastic wrapper around its newspapers. Each wrapper is 50% longer than the circumference of the rolled-up paper, and the average diameter of a paper is 5 cm. Find the number of kilometres of wrapper required to wrap the 275 000 newspapers produced daily. Give your answer correct to 2 d.p.

15

A rhombus has diagonals 24 cm and 10 cm as shown. a Calculate the length of the side of the rhombus. b Calculate the perimeter of the rhombus. Remember Pythagoras.

24 cm

10 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

16

A kite has diagonals as shown. a Calculate the length of one short side of the kite. b Calculate the length of one long side and hence the perimeter.

12 cm 3 cm

10 cm

D. COMPOSITE AREAS Figures that cannot have their areas calculated using one formula are called composite areas. The area of a composite figure can be calculated by dividing it into identifiable shapes, then adding or subtracting the area of these shapes to find the total area.

Example 1 Find the shaded area. The area is found by adding the area of the rectangle and the triangle. Total area = area of triangle + area of rectangle 1 = --- × 14 × 3 + 8 × 14 2 = 133 cm2

11 cm

8 cm

14 cm

Example 2 Find the area of this shape.

Divide the figure into two rectangles and find any unknown side lengths. Area = 18 × 8 + 7 × 8 = 200 cm2 8 cm

18 cm

7 cm 8 cm 15 cm

8 cm

18 cm

7 cm 8 cm 15 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Exercise 6D 1

Find the area of each shape (correct to 2 d.p. where necessary). All angles are right angles. a

3 cm

b

c 2 cm

5 cm

15 cm

8 cm 8 cm

4 cm

16 cm

7 cm

5 cm

9 cm 6 cm

d

10 cm

e

f 7 cm

3 cm

10 cm 11 cm 9m

15 cm

13 cm

12 m

Example 3 Find the shaded area. 5 cm

The shaded area is found by calculating the total area and then subtracting the unshaded area. Area = area large circle − area small circle = π × 72 − π × 52 cm2 = 49π − 25π cm2 = 24π cm2
75.40 cm2

2

2 cm

Find the shaded areas (correct to 2 d.p. where necessary). a

b

c 1m

3m

4 cm

10 m

8 cm

2m 4m

5m

17 m

7 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

d

e

f

23 cm

P

Q

27 cm 18 cm

8 cm 6 cm

R

S

10 cm

7 cm

11 cm

10 cm PR = 5 cm, SQ = 7 cm

3

Find the areas of the shaded regions (answer to 1 d.p.). a b

c

1 cm

4 cm

3 cm 100 m 5 cm

5 cm

d

9 cm

e

f 8 cm

8 cm 3 cm 11 cm

5 cm 17 cm 8 cm 8 cm

E. AREA APPLICATIONS This section involves practical problems using area.

Exercise 6E 1

Calculate the cost of carpeting a rectangular room 4.8 m long and 7.3 m wide, if carpet costs $72.95 a square metre.

2

The diagram shows the floor plan of a conference room. a Calculate the area of the conference room. b Calculate the cost of tiling the floor of the conference room if the tiling costs $32.80 per square metre. Scale 1 cm : 5 m

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3

A family room is in the shape of a rectangle with semicircles at either end as shown 4m in the diagram. a Calculate the area of the family room. 6m b Tiles cost $45 per square metre, and laying them costs $35 per square metre. Find the total cost of tiling the family room.

4

Concreters charge $18.90 per square metre. Calculate the cost of concreting the area shown.

8m 4.2 m 7.6 m

ro

3 -----10

of the area is used for pasture lands and the homestead. How many hectares are used for these purposes? e Calculate the value of the hobby farm if each square metre is valued at $7.20.

M

d

ad

A small triangular hobby farm is situated along a main road. a Calculate the area (in m2) of the farm. b How many hectares are there in this farm? c How many hectares are used for growing crops, if 7 ------ of the area of the farm is used for crop growing? 10

ain

5

Hobby farm

850 m

620 m

1 ha = 10 000 m2

6

A 2 m wide path is placed around a circular pond of diameter 4 m. Find the area of the path correct to the nearest whole number.

4m

2m

7

A garden bed is in the shape of a quadrant of a circle, radius 3.5 m. A path 1 m wide is to be built around the curved boundary only. Find the area of the path.

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

8

A farmer wants to spread 200 kg of superphosphate per hectare. What weight of superphosphate is required to fertilise this paddock? Give your answer in tonnes.

400 m 200 m

Example 1 A circle has the same area as a square with sides 10 cm. Find its radius.

r cm 10 cm 10 cm

Area of circle = πr 2 cm2 Area of square = 10 × 10 cm2 = 100 cm2 ∴ πr 2 = 100 100 ∴ r 2 = ---------π ∴ r 2
31.83 ∴ r
31.83 (r is positive) ∴ r
5.642 cm

9

A rectangle is 12 cm by 8 cm. a Find the area of the rectangle. b If the length of the rectangle is increased by 3 cm, find the width if the area remains the same.

10

A rectangle is 12 cm by 8 cm. If the length of the rectangle is increased by 4 cm, how must the breadth be varied so that the area remains the same?

11

A 10 cm by 16 cm rectangle has the same perimeter as a square. Which figure has the greater area? By how much?

12

A circle has the same area as a rectangle 15 cm × 7 cm. a Find the area of the rectangle. b Find the radius of the circle (to the nearest hundredth). c Which figure has the larger perimeter?

F. AREA AND PERIMETER INVOLVING SECTORS Earlier in this chapter we found the perimeters and areas of semicircles and quadrants. In this section we will find the areas and perimeters of sectors. To find the perimeter and area of sectors the fraction of the whole circle must be found first.

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Example 1 Calculate the perimeter of this sector. 30 The sector is ---------- of a circle. 360 30 ∴ The curved length = ---------- × 2 × π × r 360 30 = ---------- × 2 × π × 8 360 = 4.189 cm ∴ Perimeter = 4.189 + 8 + 8 = 20.2 cm (to 1 d.p.)

30° 8 cm

Always divide the angle by 360.

Exercise 6F 1

Find the fraction of a circle represented by these sectors. a b

4 cm

c

5 cm 12 cm 120°

60°

2

20°

Calculate the curved length and hence the perimeter of the sectors in question 1.

Example 2 Calculate the area of this sector. 50 The sector is ---------- of a circle. 360 50 ∴ Area = ---------- × πr2 360 50 = ---------- × π × (12.3)2 360 = 66.0 cm2 (to 1 d.p.)

50° 12.3 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

3

Calculate the area of the sectors in question 1.

4

Calculate the perimeter and area of these figures. a

b 5m

c

320°

100°

d

200°

57 m

115 m

10°

15.7 m

Example 3 Calculate the perimeter of this figure. 4m

70 Curved length = ---------- × 2 × π × 4 360 = 4.887 m

70°

Perimeter = 4.887 + 1.5 + 4 + 1.5 + 4 = 15.9 m (to 1 d.p.)

5

1.5 m

Calculate the perimeter of these figures. a

b 8m

5m

40° 50°

5.29 m

5.1 m 1.8 m 8.4 m

c

d 53 cm 3.7 cm 70° 18 cm 50 cm

37° 0.65 m

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Example 4 Calculate the area of this figure. 55 Area = ---------- × πr2 + lb 360 55 Area = ---------- × π × (135)2 + 81 × 135 360

55°

= 19 682.4 cm2 (to 1 d.p.)

81 cm 135 cm

6

Calculate the area of the figures in question 5.

7

Calculate the area and perimeter of this figure made of semicircles.

24 cm

8

Calculate the area of this figure.

30° 3m 50°

50°

8m 40°

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

Language in Mathematics

Johann Kepler (1571–1630) Johann Kepler was born in the German town of Wurttemberg. Although small and suffering from ill health, he was recognised as being intelligent. With a scholarship he was able to attend the University of Tubingen, where he studied first for the Lutheran ministry and then science. He studied under a master in astronomy who believed in, and taught, the Copernican theory that the Earth rotated around its own axis, and also about the Sun. Kepler taught mathematics in Graz from 1594. In 1600 he went to Prague and became assistant to Tycho Brahe, an important astronomer. After Brahe’s death, Kepler succeeded him as astronomer and mathematician to the emperor. Kepler had access to Brahe’s extensive records of observations and calculations. With his belief in the Copernican theory, he became one of the founders of modern astronomy. He developed three fundamental laws of planetary motion, now known as Kepler’s Laws, in 1609. These proposed, among other things, that the Sun was at the centre of our planetary system, and that the orbits of the planets were elliptical rather than circular. Sixty years later these laws helped Newton develop his Universal Law of Gravitation. Kepler also suggested that tides are caused by the Moon’s gravitational pull on the seas. He produced tables giving the positions of the Sun, Moon and planets, which were used for about 100 years. In 1611 he proposed an improved refracting telescope, and later suggested a reflecting telescope developed by Newton. 1

a b c d e f

How old was Kepler when he died? When and where did Kepler teach Mathematics? Describe the development of Kepler’s ideas concerning planetary motion. Research Kepler’s three laws. For how long were his tables of positions used? How are tides formed?

2

Insert the vowels in these glossary terms. a c __ rcl __ b q __ __ dr __ l __ t __r __ l c c __ mp __s __ t __ d rh __ mb __ s

☞

181

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

e k __ t __ g tr __ p __ z __ __ m

f s __ ct __ r h tr __ __ ngl __

3

Rearrange these words to form a sentence. a circle a semicircle A half is b a is of quarter quadrant A circle c are shapes calculated Composite dividing by area up the d may way than Composite more in areas one be found

4

Use every third letter to find the sentence. WDTRFHTGEHYAUJRNHEGBAVFOEDFSWAAZRDFHHJOLPMOE BQAUZDSFYOIJRBWAQAKCGIHJTIIEOPILLSGFHDEASKLAXFV BTHQHSOEYAPEFRHKOIPDNMUAECSDTCGOHNFBETWXHAUEI ODAGIBHAJKGNHODSNWEADFLTYS

Glossary area diameter perimeter rhombus triangle

circle formula quadrant sector

circumference kite quadrilateral semicircle

CHECK YOUR SKILLS 1

The formula A = 1--2- xy could be used to find the area of a: A parallelogram

2

B trapezium

C rhombus

The area of this trapezium is closest to: A 15.2 cm2 B 30.4 cm2 C 56.202 cm2 D 28.101 cm2

✓ composite figure parallelogram radius trapezium

5.1 cm

3.8 cm

2.9 cm

3

0.9 m

1.3 m

The area of this kite is: A 1.17 m2 B 2.34 m2 C 0.585 m2 D 2.2 m2

D all of these

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

4

The area of this rhombus is: A 107.01 cm2 B 13.37625 cm2 C 26.7525 cm2 D 53.505 cm2

8.7 cm 12.3 cm

5 10 m 4m

4m 20 m

6

The perimeter of this semicircular garden is closest to: A 7.2 m B 8.8 m C 4.4 m D 8.96 m

7 1.6 m

The shaded area is: A 174.9 cm2 B 99.5 cm2 C 225.1 cm2 D 300.5 cm2

2.8 m

✓

The perimeter of this shape is: A 18.05 m B 16.45 m C 10.5 m D 8.9 m

8

Tiarne has 35 m of garden edging. The radius of the circular garden she can enclose with this edging is: A 11.1 m B 5.6 m C 220 m D 3848 m

9

The perimeter of a rhombus with diagonals 5.2 cm and 8.6 cm is closest to: A 3.4 cm B 22.36 cm C 5 cm D 20.1 cm

10

The area of this figure is closest to: A 7.84 m2 B 14 m2 C 5.88 m2 D 9.8 m2

11

4.8 cm

183

1.5 cm

1.4 m

The shaded area is closest to: A 25.16 cm2 B 65.3 cm2 C 29.7 cm2 D 14.8 cm2

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

12

A farmer fertilises the paddock consisting of a rectangle and a semicircle as shown. The fertiliser is spread at the rate of 4.5 kilograms per square metre. The amount of fertiliser the farmer needs is closest to: A 10 t B 7.875 t C 12.2 t D 2.231 t

35 m

50 m

The perimeter of this sector is closest to: A 22 cm B 29 cm C 46 cm D 70 cm

13

105°

12 cm

14

The area of this sector is closest to: A 3.87 cm2 B 7.6 cm2 C 15.6 cm2 D 222.3 cm2

57° 3.9 cm

Use this diagram to answer questions 15 and 16. 15

16

4.2 m

The perimeter of the shape is closest to: A 67.42 m B 18.99 m C 14.8 m D 2.7855 m

38°

✓ 1.8 m

The area of the shape is closest to: A 5.85 m2 B 9.02 m2 2 C 13.41 m D 14.8 m2

If you have any difficulty with these questions, refer to the examples and questions in the sections listed in the table. Question Section

1–5

6–9

10, 11

12

13–16

B

C

D

E

F

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

REVIEW SET 6A 1

2

Copy and complete. a 85 cm = ___ m

b 15 000 m2 = ___ ha

Find the perimeter of these shapes. a b

c

3.5 km = ___ m

c

d 8 cm 4 cm

3m

4 cm

3 cm 10 cm

18 m

7m

3

a A rectangular field 110 m by 75 m is to be fenced. Find the total length of fencing required. b Find the perimeter of a right-angled triangle with hypotenuse length 26 cm and one other side length 10 cm.

4

Find the perimeter correct to 1 decimal place. a b 15 cm 13 cm

8 cm

5 cm 24 cm

5 cm

5

A satellite has a circular orbit 700 km above the surface of the Earth. If the radius of the Earth is 6400 km, how far does the satellite travel in one orbit?

6

Find the area of these shapes. a

b

10 m

14 cm

5m

7 cm 5 cm 15 cm

6m

7

Find the shaded area. a

b

c

40°

8 cm

8 cm 3.2 m 4.8 m

10 cm

15 cm

185

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

REVIEW SET 6B 1

2

Copy and complete. a 4.28 ha = ___ m2

b 3 cm2 = ___ mm2

Find the perimeter of these shapes. a b

c

4300 cm = ___ m

c

d 25 m

5 cm

7 cm

10 cm

15 m

12 cm

12 cm

12 m

3

a A rectangular swimming pool is 20 m by 10 m and is surrounded by a path 2 m wide. What is the perimeter around the outside edge of the path? b Find the perimeter of a rhombus with diagonals 12 cm and 16 cm.

4

Find the perimeter correct to 1 decimal place. a b

12 cm

6 cm

15 cm

5

A machine makes circular plates with circumference 60 cm. Find the diameter of the plate correct to 1 decimal place.

6

Find the area of these shapes. a

b

c

7.3 cm

22.4 cm

80°

4.2 cm 5.3 m 6.5 cm

7

18.3 cm

Find the shaded area. a 1m 10 m

15 m

b 12 cm

5 cm

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

REVIEW SET 6C 1

Calculate the perimeter of each shape. a b

7 cm

11 cm

15 cm

8 cm 5 cm

2

3

Write the formula for the area of the following shapes. a rectangle b rhombus c Calculate the area of each shape. a

b

trapezium

17 cm 3 cm

18 cm 7 cm 6 cm

25 cm

9 cm

4

Farmer Smith has a rectangular paddock that is 408 m wide and 673 m in length. Calculate the cost of fencing the paddock if fencing costs $8.53 per m.

5

Karl wishes to cut a triangle from a rectangular piece of wood, as shown. Calculate: a the area of the triangle if the base is 52 cm and the height is 64 cm b the area remaining after the triangle is removed

1.2 m

64 cm 52 cm 2.3 m

6

Find the area and perimeter of these. a b 4.5 m 75°

4.3 m

8.2 m

6m

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Perimeter and Area (Chapter 6) Syllabus reference MS5.1.1, 5.2.1

REVIEW SET 6D 1

Calculate the perimeter of each shape. a

b

12 cm 15 cm

10 cm

24 cm 6 cm 12 cm

2

3

Write the formula for the area of these shapes. a trapezium b kite

c parallelogram

Calculate the area of each shape. a

b

29 cm 16 cm

17 cm

35 cm

33 cm

4

Crystal walks around her block three times each morning. If the block is 450 m by 384 m, calculate the distance that she walks each morning. Express your answer in kilometres.

5

Deborah’s lounge room is shown opposite. Calculate the cost of carpeting the lounge room if the carpet costs $119.80 per square metre.

6

Find the perimeter of each. a

3.4 m

2.3 m

2.7 m 1.6 m

b 2.1 m

8.3 m

2.1 m

7.2 cm 4.5 m