Year 10 Mathematics General Semester Two 2013

Module 2 Measurement Name: ____________________________________ Home Group: __________________

Contents:

Set 1 Metric Units of Length.

Set 2 Perimeter.

Set 3 Area of Plane Figures.

Set 4 Area of Composite Figures.

SAC Calculator active. (Set 1 to Set 5 inclusive)

Set 5 Solids.

Set 6 Nets.

Set 7 Drawing 3-D Shapes.

Set 8 Total Surface Area

Set 9 Volume of Prisms

Set 10 Volume of Pyramids, Cones and Spheres.

SAC Calculator active. (Set 5 to Set 10 inclusive)

Metric units of length The units of length used to measure the length of a shape can be the millimetre (mm), the centimetre (cm), the metre (m) and the kilometre (km), where:

1 km = WOO m 1 m = 100 cm I cm = 1 0 mm

A simple way to convert units of length is to use the following diagram. :,,,1000 x 100 x 10

-._ --__ _____ _____ __. ___, ,--- ------1... ..------

_ _

km m cm mm .,

Ny. ,-- w,

o, __- ,__ ___ _ _ ___ - 1600 - 160 _ -:;"31

To convert from larger units to smaller units, multiply.

To convert from smaller units to larger units, divide.

Set 1 Metric Units of Length.

Measure and record the length of 10 different objects in the classroom. Choose an appropriate unit for each length measurement.

2 Fill in the gaps for each of the following: a 20 mm = cm b

5 C 130 mm = cm ci e 0.03 cm — mm f 9 0.034 m =--. cm h i 1375 mm = cm = m i k 0.08 m = mm i rt.! 670 cm .= __ m ri

13 mat = cm 1,5 cm = tam 2.8 km = 2400 mm = cm = 2.7 m cm = 6.071 km = 0.0051 km = in

3 Particle board sheets are sold in three sizes. Convert each of the measurements below into centimetres and then into metres: a 1800 mm x 900 rrun

b 2400 mm x 900 min c 2700 mm x 1200 mm.

4 A particular type of chain is sold for $2.25 per metre. What is the cost of 240 cm of this chain?

5 The standard marathon distance is 42.2 km. If a marathon race starts and finishes with one lap of a sports stadium, which is 400 m in circumference, what distance is run on the road outside the stadium?

6 Fabric is sold for $7.95 per metre. How much will 4800 mm of this fabric cost?

7 Maria needs 3 pieces of timber of lengths 2100 mm, 65 cm and 4250 mm to construe a clothes rack. a What is the total length of timber required, in metres? b How much will the timber cost at $3.80 per metre?

3

Check with your teacher. a 20 mm = 2 cm b 13 mm = 1.3 cm c 130 mrri = 13 cm d 1,5 cm = 15 mm e 0.03 cm — 0.3 mm 1 2.8 km = 2800 m g 0,034 m = 3.4 cm h 2400 mm = 240 cm = 2..4 m i 1375 mm = 137.5 cm = 1.375 m

2.7 m = 270 cm = 2700 mm k 0.08 m = 80 aim 6.071 km = 6671 m En 670 cm = 6.7 m El 0.0051 km = 5.1 m a 1800 mm x 900 mm = 180 cm x 90 cm

= 1.8 m x 0.9 m LI 2400 mm x 900 mm = 240 cm x 90 cm

= 2.4 m x 0.9 m. c 2700 mm x 1200 mm = 270 cm 120 cm

= 2_7 in x 1.2 m

4 5 6 7

$5.40 41 400 in or 41_4 km $38.16 a 7m b $26.60

Perimeter The perimeter of a shape is the total distance around that shape.

1. The perimeter of a shape is. the total distance around that shape. 2. For circular figures the term circumference is used instead of perimeter. 3.

'Shape . ' . Square Rectangle Circle

. . I I 1 111 - . - . . ,- . ..

11 .. ,, ..

Formula for P = 41, where / is the side P = 2(/ + w), where 1 is P = Circtwilerence calculating length the length and w is the C = 27-cr, where r is the perimeter width radius ... .. - As 2r = (/ (dianteter),

C = iCti

4. When finding the perimeter of a shape, make sure that all measurements have the same units.

txample Find the perimeter of each of the following shapes. (Where appropriate, state your answer correct to 2 decimal places.) a 12 cm b c

CM 5 cm

a P=5 +12+5 +12 = 34 cm

C 27rr

r = 7

C=2 x7rx 7 = 43.98 cm

P=6 x5 = 30 cm

= 34 cm

txampie Find the perimeter of each of the following shapes. a 8 cm

6 cm 7

a Curved length C

I (7r x 6) 2 -1

= x 8.84 2

9.42 cm P=9.42 +8 +6 +8

= 31.42 cm

b Curved length = x C

x 27rr 360

= 12" X 2 x x 7 360

----- 14.66 cm P= 14.66 + 7 + 7

= 28.66 cm

txample

A rectangular paddock 70 m by 48 m needs to be fenced with four rows of wire. What is the total length of wire required to complete the fencing?

P = 2(1 + w)

1 = 70, w = 48

P = 2(70 + 48) = 2 x 118 = 236 m

Length of wire required = 4 x P =4 x 236 = 944 m

cm 8 cm

5 cm 6 cm

12 cm

( 30 cm

14 cm

30 cm

21 cmn

100 i n

14 cm

Set 2 Perimeter

Where appropriate, give answers correct to 2 decimal places. 1 Find the perimeter of each of the following shapes.

a 10 cm b 15 mm

2 cm h 2c111

1 cm

1 cm

2.5 cm

3 cm

4 cm

2 Find the perimeter of each of the following: a a rectangle 20 cm by 12 cm b a regular hexagon of a side length 15 mm c an equilateral triangle of a side length 12 cm d a circle of diameter 25 cm e an isosceles triangle with its base 12 mm and equal sides of length 16 mm each.

3 Find the perimeter of each of the following shapes. (Give your answer correct to 2 decimal places.) a b - 10 cm

cm-o- cm

l a

2a

3a

26 cm 15 cm 64 cm 78.5 cm 77.12 cm 95.99 cm

et A r1 g C111

4 a 148.5 cm 5 650 m

9 to cm

8 cm

-4-15 cm 60

t

128 mm

4 Find the perimeter of the shapes below.

a 35 cm 14 cm

- 7 cm

-4-8cm 2cm -4-8cm--).-

15 c

5 A rectangular paddock 38 m by 27 m is fenced with 5 rows of wire. What is the total length of wire needed?

8 The length of a rectangular pool is twice its width. If the perimeter of the pool is 81 m, find its dimensions.1

9 A length of masking tape, 100 cm long, is wrapped around a rectangular block along the edges once. How long is the block if its width is 15 cm?

10 A circle has a circumference of 81.64 cm. What is its radius?

Perimeter b 40 mm c 56.55 cm d 24 cm f 31 cm g 10 cm h 30 cm

b 90 mm c 36 cm e 44 cm b 74.99 cm c 43.56 cm e 262.83 m f 71.98 cm h 174.55 cm i 163.98 cm b 47.14 cm c 54.27 cm

8 27 m x 13.5 m 9 35 m

10 13 cm 11 406.28 m, 412.57 m, 418.85 m 12 12 cm

Area Area is the measurement of the amount of space inside a flat shape. We measure area in square units, such as mm2, cm2, m2 and km2. This can be done by counting squares s

inside a flat shape. For instance, the area of the rectangle below is found by counting the number of squares (each 1 cm 2) inside the shape.

Hence, the area of the rectangle is 6 cm 2 .

The area units are based on standard squares of different sizes; the common ones being the:

square millimetre — a 1 mm x 1 mm square — (1 mm 2 ) square centimetre — a 1 cm x 1 cm square — (1 cm 2 ) square metre -- a lmx 1 m square — (1 m 2 ) hec tare — a 100 in x 100 in square —(1 ha = 10 000 m 2 )

Many simple geometric shapes have a formula that allows area to be quickly calculated.

These are to be remembered.

Shape Formula

1 . Square

1

A = 1 2 , where 1 is a side length.

2. Rectangle A = / x w, where / is the length and w is the width.

3. Triangle A bh, where b is the base length and h the height.

4. Trapezium a A = (a + b) x h, where a and b are lengths of parallel sides and h the height.

12

Shape

Formula

5. Circlc A = zr2 , where r is the radius.

6. Parallelogram A = bh, where b is the base length and h the height.

7. Sector A =

360° x Kr, where 0 is the sector angle

in degrees and r is the radius.

8. Rhombus A = xy, where x and y are diagonals.

Heron's formula

If the lengths of all three sides of a triangle are known, its area, A, can be found by using Heron's formula:

A = Vs(s — a)(s — b)(s — c) where a, b and c, are the lengths of the three sides

and s is the semi-perimeter or s = a + b + c 2 e

The area of an ellipse can be found using the formula:

A = gab, where a and b are the lengths of the semi-major and semi-minor axes.

9.2 cm

6 cm

(e)

(g)

(0

10.5 cm

10 cm

13 cm

20 cm

(h)

2 cm

Set 3 Area of Plane Figures Find the area of each of the following figures:

1. (a) square of side 6 cm (c) square of side 8.5 m (e) square of side 1.1 cm (g) rectangle 6 km by 4 km (it) rectangle 3.2 cm by 7.5 cm (k) parallelogram base 4 cm height 10 cm (m) parallelogram base 6.7 cm height 3.1 cm (o) parallelogram base 10.5 m height 4.6 m (q) rhombus of diagonals 3 cm and 5 cm (s) rhombus of diagonals 7 cm and 9.5 cm (u) rhombus of diagonals 3,2 cm and 6.1 cm

2. Find the area of

(b) 9 cm

14 cm

(c)

(d)

4 cm

5 cm

16cm

1.5 cm

5 cm 8 cm

(a)

12 cm

3. Find the area of each of the following circles n ( . 22 7 -

(a) radius 7 cm (c) radius 3.5 cm (e) radius 10 m (g) diameter 10 cm (1) diameter 42 cm

4. A rectangle of length 6 cm has area of 24 sq cm. Find its breadth. 5. A rectangle of breadth 8.5 cm has area of 102.0 sq cm. Find its length. 6. A square has area of 121 sq cm. What are its dimensions? 7. What are the dimensions of a square whose area is 133 sq cm`? 8. Find the height of a parallelogram whose base is 9 cm and area 87.3 sq cm.

I. (a) 36 cm 2 (c) 72.25 m 2

(e) L21 cm2 (g) 24 km 2

(1) 24 cm2 (k) 40 cfn. 2 (rn) 20.77 cm 2

(o) 48.3 m 2

(q) 7.5 cm 2 (s) 33.25 cm" (u) 9.76 cm 2

2. (a) 36 CM 2 (b) (); in ) (c) 40 CM 2

(d) 3 cm 2

(e) 29.75 cm 2 (1) 60 cm 2 (f) 48.3 CM 2 (1) 30 cm 2 (g) 85 cm 2 (h) 45 cm 2

3. (a) 154 cm 2

(c) 38.5 cm 2 (e) 314.3m 2 (g). 78.57 cm' (h) 12.57 cni 2

4. 4 cm 5. 12 cm 6. 11 cm side 7. 11,53 cm side - 8. 9.7 cm

Composite areas

The area of a composite figure can be calculated by considering it as the sum or difference of simpler shapes.

Examples: 1. Find the area of the swimming pool shown. The pool can be thought of as a rectangle . , . .

I +10-6 m and two semicircles which combine to + , ' make a full circle.

1-•-- 204m

Area rectangle = LW

= 204x 10-6

= 21624m 2

Area circle (r = 5-3 m) = .rrr2

= 3-142 x (5-3) 2

= 88-247 m 2

Total area = 216.24 + S8.247 = 304.487 m2

30 cm

4 cm

12 cm :10 cm

16 cm

25

30 cm

3 mm

2 mm 2 mm

4

nm

Set 4 Areas — composite shapes

1 Find the area of each of the following shapes.

a 52 Mil b 10 cm

12cm 21 rim -------1 23 cm 30 ram 11 cm

35 cm 19 mm

in

20 cm /9 cm

8 cm : 0cm 20 cm

40 cm

cm 15 cm

21 mm

37 mm

15 mm

Z

15 cm

8 cm

18 mm

1

4-- 30 cm

3 Find the shaded area in each of the following. a b r----

9 cm

50 cm

I

40 cm )20 cm

4 What is the radius of a circle with an area equal to that of a square 8 m wide?

5 The Department of Roads has a machine that lays bitumen road at a rate of 20 linear metres every 15 minutes. The bitumen is 6 m wide. The machine works for 7 hours per day, and the Department must complete a road 12.5 km long.

i How long will it take to complete the road? ii What will be the total area of bitumen laid by the machine?

I 6 Find the area of each piece of this

tangram puzzle. 5 cm

7 The tables in Joshua's classroom are trapezoidal in shape. Each table needs a new sheet of veneer on its surface. If there are thirteen tables in the classroom, what area of veneer will be required?

85cm

a 1263 mm 2 b 583 cm' c 942.48 cm2 d 13.72 m2 e 273.10 cm 2 f 349.94 mm2 g 655 cm= h 157.5 cm' i 25.13 mrn2 j 47.12 cm= k 32.47 crn 2 1 25.63 mm2 m 294 cm2 n 437.5 cm'

3 a 3063.05 cm2 b 76.37 cm2 c 48.29 cm' d 69.53 rnm2 e 89.43 m2 f 422.43 mm2 g 27.97 m= h 18.27 mm= i 123.61 cm2 j 52.15 cm2 k 2194.29 cm' 1 441.06 cm"

4 4.51 m Si 156.25 hours 22.32 days ii 75 000 m2 6 A, B, F- 12.25 crn2 ; C, G-6.125 cm2 ;

D, E---24.5 cm' 7 90837.5 cm2

sphere

Solids

As well as plane figures, technical drawing is very concerned with solids. Solids are 3 dimensional figures. You must learn to recognise and name these.

rectangular prism

hexagonal prism

triangular pyramid

pentagona pyramid

cone cylinder

Set 5

1. From the solids above, choose the one whose shape is closest to each of the following objects:

a block of ice b a chalkboard duster c a tin of peaches

d a basketball e a dunce's cap

f this textbook

g an ice cream cone h a tin of mushrooms

i a tennis ball

j The Great Pyramids of Egypt

Nets (Ask your teacher to photocopy each net)

The faces forming a solid can be drawn as plane shapes, which are joined across the edges to form the solid. The complete set of faces forming a solid is called its net. Note that for some figures, different nets can be drawn.

Set 6 Cut out... fold the black lines...

Tetrahedron

Cut out... fold the black lines...

Cut out... fold the black lines...

Pyramid

Cone

Drawing 3-D shapes The worked example below shows step-by-step instructions that will help you to draw a rectangular prism, a right-angled wedge and a square-based pyramid.

Construct each of the following shapes. a A rectangular prism whose dimensions are 5 cm x 3 cm x 4 cm b A right-angled wedge with base 5 cm x 4 cm and height (at the tallest end) 3 cm c A square-based pyramid with the length of the base 5 cm and the height 6 cm

CID DRAW

a I Draw a rectangle ABCD. a 13 Note: Figures reduced to fit.

1)

2 Draw a rectangle EFGH. It must be congruent (have the same shape and size) to rectangle ABCD and overlap it at the top right corner.

3 Join the corresponding vertices of the two rectangles with straight lines. That is, join A to E, B to F, C to G and D to H. Write in the given dimensions (length, width and height).

b I Draw a parallelogram ABCD.

II

4 cm

C

.\

2 On the side DC of the parallelogram draw a rectangle DCFE. (That is, DC is the shared side, belonging to both the parallelogram and the rectangle.)

3 Join A to E and B to F with straight lines. Write in the dimensions of the wedge that are given.

: 1 Draw a parallelogram ABCD. C 1) (

A 1-3

2 Draw the diagonals of the parallelogram (AC and BD). From the point of intersection of the diagonals draw a vertical line 6 cm long, which will represent the height of the pyramid. Call the end-point of the line E

3 Join point E to points A, B, C and D with straight lines. Write in the dimensions of the pyramid that are given.

A

Set 7 Drawing 3-D Shapes 5 cm

Try these Construct each of the following shapes on a separate sheet of paper.

1 A rectangular prism whose dimensions are 4 cm x 3 cm x 2 cm.

2 A rectangular prism whose dimensions are 3 cm x 2 cm x 6 cm.

3 A cube with side length 3 cm.

4 A right-angled wedge with base 4 cm x 3 cm and height (at the tallest end) 5 cm.

5 A right-angled wedge with base 6 cm x 7 cm and height (at the tallest end) 3.5 cm.

6 A right-angled wedge with a square base 4 cm x 4 cm and height (at the tallest end) 2.4 cm.

7 A square-based pyramid with the length of the base 4 cm and the height 7 cm.

8 A square-based pyramid with the length of the base 3.5 cm and the height 5 cm.

9 A rectangular-based pyramid with base 4 cm x 6 cm and height 8 cm.

4

A 4 cm R

A 6 cm

- ANSWERS Drawing 3-D shapes

Note: Answers not to exact size.

1 F G 2

AF IW G Alli ril R B

2 cm P .rcil l

6 cm H

A 4 cm n

F 6 E F

3.5 cm

■ 1 I 1 I I I I I I M i 4 cm C A .4 cm R

9 7 8

A.,„ 3.5 cm

A 3-5 cm

I ) AllarCM

A 4 cm

_ \' 6 cm

A 4 cm

Total Surface Area The total surface area of a three dimensional object is the sum of the areas of each surface.

(a) The Cube

A cube has 6 square faces of equal area. In a cube of side / cm, each face has area 1 2 cm 2 . Total siAice area = 6/ 2 cm 2

Example Find the total surface area of a cube of side 3 cm.

Solution T.S.A. = 612

= 6 x (3) 2 = 6 x 9 = 54 CM

2

T.S.A. = 54 cm2

(b) The Cuboid A cuboid has six rectangular faces with each pair of opposite faces being of equal area. Area of top = area of base

=1 x b Area of front = area of back

= 1 x h Area of each end =h x h

Fig 7.10

Total surface area = 2 (area of top + area of front + area of end) = 2(1xb+Ixh-i-hxh)

Example Find the total surface area of a cuboid of side 10 cm by 6 cm by 4 cm.

Solution T.S.A. = 2(10 x 6 + 10 x 4 + 6 x 4)

-'11111111111111111111111111111P. 4 cm = 2(60 + 40 + 24)

I = 2(124) )-- — — — = 248 cm 2

---

--- 6 1-TTI T.S.A. = 248 cm 2 10 cm

(e) The Cylinder

41M

Total surface area = area of curved surface + a both ends.

If the curved surface were split lengthwise and unr( it would be a rectangle of length equal to the circum-ference of the cylinder and width, the height of the cylinder.

.'. area of curved surface = 2mrh Area of each end = nr 2 T.S.A. = 2nrh + 27rr 2

=- 2nr(r. + h)

Example Find the area of a cylinder of radius 7 cm and height 10 cm.

Solution T.S.A. = 2nr(r + h)

= 2 x —22

x + 10)

= 2 x 22 x 17 10 L m = 748 cm 2

I .S.A. = 748 cm2.

Summary of the rules TSA area of 6 squares

= 6,x 2

TSA — Cube

Rectangular prism

Cylinder

Cone

Sphere

TSA = area of 6 rectangles = 2 (Iw+ + th)

TSA = area of 2 circles and area of curved surface = 27c1-2 + 27crh = 2nr(r+h)

TSA = area of circle and area of curved surface ?Tr

2 + ?Er's

= nr(r+s)

TSA = 4nr2

Square-based pyramid TSA = area of the = +4 square base and area of 4 triangles

b2 x-lbh = 132 + 2bh

1 \Z5 cm

75-6i1\

Set 8 Total surface area (TSA)

1 Find the TSA of each of these solids.

a 11,3 cm

d 6Aw

1111 10 cm

10 cm 3 cm

12

h 9 6 cm

8 cm

i 18 crn

1 12 cm

12 cm n I I

1 35 cm

1111r20 an I r

L a 3402.34 Trim" b 1592.79 rnm2 c 68.77m2 d 159956 cm' e 3619.11 cm' f 2462.22 mmz g 9424.78 cm2 h 1413.72 cm2

Volume of a prism

In general,

Volume of a prism Area of base x height and is measured in cubic units.

Find the volume of each of the following prisms:

(a)

5cm

71

Volume Area of base x height = x W) x height = (2 cm x 2 cm) x height = 4 cm 2 x 5 cm = 20 cm 3

2em

(b) A triangular prism is a prism with a uniform triangular cross-section.

Thus the base of the triangular prism is the face shaded blue, since the base of a prism must be the same as the uniform cross-section.

Thus, the volume of the prism

2m

= Area of base x height = Area of a triangle x height =( x base x height) x height =( x 2 m x 2 m) x height = 2 m 2 x 3 m = 6 m 3 .

Find the volume of each of the following prisms:

(c)

I

Volume = Area of base x height = Area of circle x height

/Ocm = nr 2 x height

/

= 2772- x 49 x height = 22 x 7 cm 2 x height = 154 cm 2 x 10 cm = 1540 cm'

- i..irJ iii

The area of the base of this prism has been given.

1 Volume = Area of base x height = 15 m 2 x 8 m

8cm = 120m 3

9cm

5cm

4.

7m

3m 6.

4cm

71

6m

9. 1 0 .

Find the volume of each of the following prisms: 1. 2.

3.

5

4cm

6cm

7. 8.

9cm

22cm i

5cm

Set 9

12cm

9m

using it = 3.142

/Ocm

„

20cm

1 3. 14.

15cm

18cm

2cm

it = 3.142 Tr = 3.142

15. 16.

23cm 8cm

10.

71 =

17. 18.

19. 20.

I. 125 cm 3 2. 480 cm 3 3. 63 m 3 4. 486 M 3 5. 64 cm 3 10.1188 cm 3 11.31.42 cm 3 12. 12,320 cm 3 13.1696.68 cm 3 14 18. 96m 3 19. 144 cm 3 20. 216 cm3

6.108 cm 3 7.126 cm 3 8. 3 1 0 cm 3 9.225m 226.224 cm 3 13. 308 cm 3 16 7969.5 cm 3 17. 280 cm 3

base

Area of base =A

Volume of pyramids • Pyramids (including cones) are not prisms as the cross -section changes

from the base upwards.

• It has been found that the volume of a pyramid is one-third the volume of an equivalent prism with the same base area and height.

Volume of a pyramid = AH

• Since a cone is a pyramid with a circular cross-section, the volume of a cone is one-third the volume of a cylinder with the same base area and height.

Volume of a cone = -13 AH

= -3 1rr`h

111,1•1:4 :44 4 Dia ;AI

Find the volume 1 a V = -3 /17— h

r = 8, h = 10

V=xrcx 8 2 x 10 = 670.21 cm3

Volume of spheres • Volume of a sphere of radius, r, can be calculated using the formula: V= lirr 3 .

Find the volume of a sphere of radius 9 cm. Answer correct to 1 decimal place.

Vi1 = X 71- X93 3

= 3053.6 cm3

3 cm 4.2 m

12

15 cm ;

20 cm

3 cm

4.2 cm

:111

15 cm

2 Calculate the volume of each of these solids. a

18 mm

24 cm

6 Find the volume of each of the following pyramids.

a

30 cm —Volume

1 a 27 cm3 c 3600 cm3

2 a 450 mm3 3 a 6333.5 cm3

c 280 cm 3 4 a 7.2m3

c 1436.8 mm3 5 a 377.0 cm 3 6 a 400cm3

b 74.088m3 d 94.5 cm3 b 360 cm2 b 19.1m 3 d 288 mm 3 b 14 137.2 cm3 d 523 598.8 cm3 b 2303.8 mm 3

b 10 080 cm 3 c 576 cm3

Volume 1 Find the volumes of the following prisms.

a

[Base area: 25 mm2] [Base area: 24 cm 2]

3 Find the volume of each of the following. Give each answer correct to 1 decimal place. where appropriate. a

41111. 4.110 t 14 cm 2.7m

-

7c

12 mm

8 mm

6 mm

4 Find the volume of a sphere (correct to 1 decimal place) with a radius of: a 1 .2 m b 15cm c 7 mm d 50cm.

5 Find the volume of each of the following cones, correct to 1 decimal place. a

;

10 cm.

_

20 mmi

6 cm