Chapter 6Measurement What you will learn Australian curriculum 6A 6B 6C 6D 6E 6F 6G 6H 6I Review of length (Consolidating) Pythagoras theorem Area (Consolidating) Surface area prisms and cylinders Surface area pyramids and cones (10A) Volume prisms and cylinders Volume pyramids and cones (10A) Spheres (10A) Limits of accuracy (Extending) MEASUREMENT AND GEOMETRY Using units of measurement Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids. (10A) Solve problems involving surface area and volume of right pyramids, right cones, spheres and related composite solids. Pythagoras and trigonometry Solve right-angled triangle problems. (10A) Apply Pythagoras theorem to solve three-dimensional problems in right-angled triangles.
Online resources Chapter pre-test Videos of all worked examples Interactive widgets Interactive walkthroughs Downloadable HOTsheets Access to HOTmaths Australian Curriculum courses Monolithic domes Monolithic domes are round one-piece structures with a smooth spherical-like surface. They offer excellent protection from earthquakes, bushfires and cyclones because of their shape and durability. They are extremely energy efficient because of the minimal surface area for the volume contained within the structure. The first type of monolithic domes used were igloos, which are very strong and provide good insulation in freezing conditions. The minimal surface area of the dome means that there is less surface for heat to be transferred to the outside air. A cube, for example, containing the same volume of air as the dome has about 30% more surface area exposed to the outside air. Volumes and surface areas of spheres and other solids can be calculated using special formulas.
10 cm 100 1000 100 10 km m cm mm 1000 100 10 radius (r) diameter (d) θ r
2.1 m 4.5 m 3.4 m x m r cm
3 cm 240 ( ) 40 105 30 220 m 0.5 km 34 cm 12 cm 32 cm 185 m 2.6 km
6.2 cm x cm 12 cm 10.4 cm 8.4 m x m 7 cm 1.2 m 28.4 mm 1.1 km r m 4 m 2 m 60 1 km 60 6 cm 80 205 5 mm 3 cm
2 7 5 Perimeter = 17 x x 5.3 Perimeter = 22.9 2x Perimeter = 0.072 2.8 2x 11.61 1.5x 3.72 x 3x 2x 7.89 Perimeter = 16.2 Perimeter = 10.8 Perimeter = 46.44 7.9 3.6 1.5 0.3 3.6 10 2.2 6 12
3 m 5 m 2.5 m 1 km x
l w rotating circle fixed circle r r
( ) b a 1 a 2 c 3 c ( b ) c b a
1.1 m x cm 5 cm y m 1.5 m 9 cm
D A E H C B G F E 4 A 7 B H 2 E 65 B y d c x z b a x
10 m 7 km 3 cm 5 m 15 km 4 cm 0.2 mm 0.37 km 0.21 km 1.8 mm 72.1 cm 27.3 cm 0.3 m 12 m 2 m 5 m 0.7 m 9 m 1.32 cm 0.71 cm 19.3 cm D A H E 5 H C B G 3 G 2 F 0.14 cm 24.2 cm 0.11 cm H D G C 14 E A 8 F 4 B H G E F D 3 C D C 9 7 A 1 B A E 8 B 5 F
3 8 2 12 13 6 4 11 5 10 6 12 12 9 15 1 5 2 B 3 cm 1.2 m 1.7 cm 35 m A 3.5 cm A 49 m 2.6 m B 1.9 cm 26 m A B 5.3 cm A B 0.5 km 5.3 cm A B 2.1 km 1.8 km 17.2 km 12.3 km 19.7 km A B
x mm x m 9.3 mm 3.7 m x cm 5 cm 15.5 m 11.4 mm 3 cm 1.5 m x m 3 m 2 m 6.2 km 6.7 km x km 2.93 cm 5.31 cm x cm 4.04 cm 8.2 km 8 6 4 8 6 4 2 2 O 2 4 6 8 y 2 4 6 8 x 2 cm 1 cm r cm
A 3 m E 4.5 m B D F H 3.5 m C G
3 m 2 m Area = 6 m 2 1000 2 100 2 10 2 km 2 m 2 cm 2 mm 2 1000 2 100 2 10 2 1 m = 100 cm 1 m 2 1 m = 100 cm
l w h l b a x y x h h y b b θ r 1 cm 2 10 mm 10 mm 2 2 2 km m cm 1000 2 100 2 100 2 = 10 000 and 1000 2 = 1 000 000
3 cm 3.3 m 1.06 km 2 cm 5 cm 5.8 m 1.3 m 2.3 mm Area = 11 mm 2 l mm Area = 0.5 m 2 a m 0.4 m
( ) 80 3 cm 5 m
5.2 m 1.3 km 5 cm 2.8 km 10.5 m 0.3 mm 0.1 mm 0.2 mm 7 m 10 cm 15 cm 64 m 23 m 0.4 mm 2 3 km 0.25 mm 1 4 km 3 km 0.82 m 10 cm 1.37 m 2 cm 6 cm
5.2 cm 1.8 km w cm l m h km 2.8 m 1.3 m 5.34 mm h mm 18 m a m x m Area = 80 m2 r cm d m d m 10 m 6 cm 120 7 m 100 2 m 6 km 150 3 mm 325
8.2 m 6.5 m 7 m 4 m 1.7 m 1.8 m 5 cm 10 m 1.6 m 28 km 18 km 0.25 m 0.3 m 4.2 mm 26 km 0.7 m
a h b x x x x
r r h 2πr h r
common face 3 cm 5.3 m 5 cm 8 cm 1.7 m 5 cm 3 cm 3 cm 8 cm 5 cm 1.7 m 2πr 5.3 m 1.7 m
6 cm 8 cm 10 cm 20 cm 4 cm 10 cm 2 cm 3 cm 2 cm 1 cm
3 cm 2.1 cm 3 cm 6 cm 2.7 mm 5.1 mm 1.3 mm 4.5 cm 7 cm 15 m 8 m 20 m 9.2 m 12.8 m 5.2 m 6.4 m 17 m 5 mm 7 mm 3 mm 1.2 cm 4 mm 25 m 10 m 12 mm 6 mm 8.66 m 16 mm 5 mm 0.5 cm 1.2 cm 1.5 cm 1.4 m 2.1 m 0.76 cm 0.8 cm 1.9 m 2.3 m 4.8 m
1.6 m 3 cm 4 m 10 cm 2 m 20 cm 6 m 40 40 cm 1.8 3.2 7 2 7 1.4 8 10 3 8 10 33 5.2 1.8 26 4 cm 4 cm 8 m 2 cm 6 cm 10 cm 10 m 20 cm 20 m 40 m 15 m
2 m 5 m c a b x h d r h
r r x y h r r θ h
A s θ s r B s r s apex s r right cone r 2πr
apex right squarebased pyramid 2 m 4.5 m 22 mm 25 mm
( ) ( ) ( ) h s r 5 m 5 cm 6 cm 14 m 10 cm 2 cm 2 cm 2 cm 4 cm 3 cm
9 mm 0.5 m 11 km 12 mm 0.8 m 15 km 8 cm 6 m 0.2 m 0.3 m 4 m 5 cm 1.1 cm 26 mm 10 m 1.5 cm 48 mm 2 m 6 m 4 m
3 cm 4 cm 2 cm 5 cm 3 cm 4.33 cm 10 m 4.33 cm 8 cm 3 m 5 cm 8 m 9 mm 5 m 6 mm 10 mm 2 m 7 m
10 cm 10 cm 8 cm 20 cm
11 cm 5 cm 7 cm 4 cm 13 cm 3 cm 8 cm x cm 13 cm 12 cm x cm 9 cm 40 cm 5 cm 8 cm 12 cm x cm 4 mm 17.6 mm 5 cm 80 5 cm 4 cm 210
4 cm 7 cm 6 cm 10 cm 1.1 m 5 cm 50 cm 8 cm 3 cm 12 cm 8 cm 7.2 cm 6 cm 120 7.5 cm O
10 10 10 3 6 3 6 3 6
1000 3 km 3 m 3 ML 1000 3 1000 1000 kl 100 3 100 3 1000 1000 cm 3 L 10 3 10 3 1000 1000 mm 3 ml l w h h x x r h 2 cm 4 m 2 m 3 m 6 cm
3 cm 8 cm 5 cm 2 cm 2 cm 6 cm
10 cm 2 m 12 mm 2 cm 4 cm 5 cm 30 m 35 m 10 m 7 mm 3.5 mm 10.6 mm 2 cm 10 m 1.5 cm 12 m 14 m 5 m
5 m 2 cm 3 cm 20 m 1.2 mm 2.4 mm 1.7 mm 3 m 8 cm 2 km 10 km 3.5 cm 7 m 2 m 3 m 3 km 7 cm 21.2 cm 5 cm 2 2 cm 0.38 m 18.3 cm 0.12 m 24.5 cm 0.5 m 12 cm 6 m 2 m 0.8 m 3 m 12 cm 2 m
5 cm 5 m 2 cm 8 cm 5 m 1.1 m 1.6 m 2.3 m 4 cm 2.6 m 5 mm 5 cm 11 mm 8 mm 4 cm 3 cm 9 cm 6 cm 5 4 3
r θ h 4 m 10 m 10 m 1 m 1 m
h h h x x r r 23 mm 1.3 m 1.4 m 29 mm 1.2 m
6 m 4 cm 7 mm A = 5 m 2 A = 2 cm 2 A = 25 mm 2 3 cm 5 km 18 m 12 km 2 cm 15 m 5 km 13 m 2 cm 4 cm 3 cm 1.4 mm 0.6 mm Perpendicular height = 2 cm Perpendicular height = 4 cm Perpendicular height = 1.2 mm
1.6 mm 20 m 2.6 m 3.5 mm 60 m 1.1 m 0.6 m 20 cm 6 m 5 cm 1.3 m 2 m 6 m 5 cm 8 m 3 m 6 cm 4 m 2 m 6 m 3 m 4.8 mm 1.5 mm 5.2 mm
3.96 m 2.6 m 10 m 2.8 m 7.1 m 3 cm ice-cream 6 cm depth of ice-cream
r 2 h 2 h 1 r 1
r
r m 5 cm 7 cm 5 cm 2 cm s cm
( ) 0.5 m 2 cm 38 mm 1.36 m 18 cm 0.92 km
r cm r cm r km r m r cm r mm
5 cm 4 cm 1 m 2 m 4 m 5 m 0.7 cm 1.3 cm 8 mm 4 mm 3 m 5 m 1 m 3 cm 2 cm 2 m 2 cm
1 m 13 m 3 m 19 m 28 cm 2 cm 20 cm hollow 2 m 2 m 2 m 1 cm 10 cm 15 cm
x x x r
85.5 86.5 85 86.0 87
85.95 86.05 85.8 85.9 86.0 86.1 86.2
2 4 6 3 9 6 2 m 6 m 10 m
1 m 6 cm 4 m 1 m 2 m 2 m 10 cm 5 cm 1 cm A C A C B
6 cm 10 cm 12 cm 5 cm 8 cm 2 cm
Up for a challenge? If you get stuck on a question, check out the 'Working with unfamiliar problems' poster at the end of the book to help you. 45
Area Units of area 1000 2 1002 10 2 km 2 m 2 1000 2 100 2 10 2 Formulas circle square rectangle triangle r l w h A = pr 2 A = l 2 l b 1 A = lw A = bh 2 kite sector rhombus parallelogram trapezium x a x y r y h h θ b b θ 1 1 A = pr 2 1 A = xy A = bh A = (a + b) h A = xy 360 2 2 2 For prisms and pyramids draw the net and add the areas of all the faces. e.g. Surface area TSA = 4 triangles + square base Cylinder r TSA = 2prh + 2pr 2 curved base + surface ends Cone (10A) h TSA = prs + pr 2 curved + base r s 2pr Sphere r TSA = 4pr 2 (10A) For composite solids consider which surfaces are exposed. h Measurement Limits of accuracy Usually ± 0.5 the smallest unit of measurement e.g. 72 cm is 71.5 cm to 72.5 cm. Pythagoras theorem Volume Units of volume 1000 3 1003 10 3 km 3 m 3 cm3 mm3 1000 3 100 3 10 3 Units of capacity 1000 1000 1000 megalitres (ML) kilolitres (kl) litres (L) millilitres (ml) 1000 1000 1000 1 cm 3 = 1 ml l V = lwh b c c 2 = a 2 + b 2 a Can occur in 3D shapes Length Units of length 1000 100 10 km m cm mm Perimeter is the distance around the outside of a closed shape. Circumference of a circle C = 2pr = pd Perimeter of a sector r θ P = 2r + 2pr 360 θ h w 1000 100 10 For right and oblique prisms and cylinders V = Ah, where A is the area of the base h is the perpendicular height. rectangular prism cylinder h r V = pr 2 h For pyramids and cones: (10A) 1 V = Ah, where A is the area of the base. 3 Cone: 1 V = pr 2 h 3 h 4 Sphere: V = pr 3 (10A) r 3
10.4 cm x cm 7.1 cm 5 m 100 7 km x km 12 10 7 m 12 m
10 cm 9 cm 4 cm 8 m 9 m 4 m 80 6 cm 8 m 7 m
6.2 m 12 m 13 m D A 4 C B H E 7 m 11 m 7 F G 2 10 cm 5.2 m 18 m 6.4 m 13 m 8.1 m 4.2 m 7 m 9.8 m w m deck 9.2 m 5.3 m 6 m 4 cm 5 cm 8 m 5 m 6 cm 10 cm 8 cm 10A 12 cm 13 cm 20 cm 10 cm 10 cm 10A 10A 10 cm 4 m 12 cm 13 cm
5 m 3 cm 3 cm 4 m 0.9 mm 7 m 10 cm 2.5 mm 1.6 mm 3.9 cm 5 cm 3 cm 5 cm 2 cm 3.2 cm 4 cm 1.8 cm
36 cm 39 cm 51 cm c m 2 m 8 m 1 m 6 m