measurement - wordpress.com · measurement and geometry using units of measurement solve problems...
TRANSCRIPT
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Chapter
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)
6A
6B6C6D
6E
6F
6G
6H6I
M E A S U R E M E N T A N D G E O M E T R Y
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.
Solve right-angled triangle problems.(10A) Apply Pythagoras’ theorem to solve three-dimensional problems in right-angled triangles.
What you will learn
Australian curriculum
6 Measurement
Pythagoras and trigonometry
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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.
Chapter pre-testVideos of all worked examplesInteractive widgetsInteractive walkthroughsDownloadable HOTsheetsAccess to HOTmaths Australian Curriculum courses
Online resources
Monolithic domes
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••••
100°
10 cm
× 1000
÷ 1000
km m
× 100
÷ 100
cm
× 10
÷ 10
mm
•
radi
us (r
)
diameter (d)
• θ r
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3.4 m
2.1 m
4.5 m
x m
∴
∴
r cm
∴
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3 cm240°
( )
40° 105°30°
185 m
220 m
0.5 km
2.6 km
12 cm
32 cm34 cm
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6.2 cm
10.4 cm
12 cm
x cm
8.4 m
x m
7 cm 1.2 m
28.4 mm
1.1 km
r m
4 m
2 m
1 km60°
6 cm60°
5 mm
80°
3 cm
205°
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5
2
7
Perimeter = 17
x
5.3Perimeter = 22.9
x2x
Perimeter = 0.072
2.8 2x
Perimeter = 16.2
11.61
3.72
7.893x
Perimeter = 46.44
1.5x
2xx
Perimeter = 10.8
1.5
3.6
7.9
0.3
10 2.2
3.6
6 12
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3 m5 m
2.5 m 1 km
x
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l
w
r
rrotatingcircle
fixedcircle
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( )
b
a
b
c
c1
a
3
2
•( )
•••
a
b
c
√
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9 cm
5 cmx cm
1.5 m
1.1 m
y m
∴
∴√
∴√
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D E F
GH
C
A B
A
E
B
4
7
∴
∴√
E B
H
2
√65
√
∴√
√√ √ √
zx
y
ba
d
x
c
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4 cm
3 cm5 m
10 m
15 km
7 km
0.2 mm
1.8 mm
0.21 km0.37 km
72.1 cm
27.3 cm
2 m 5 m
9 m
12 m
0.3 m
0.7 m
0.71 cm
1.32 cm
19.3 cm24.2 cm
0.11 cm
0.14 cm
D
A
CG
B
F2
35
H
E
DH
GC
BF
EA
84
14
C
GH
E FD
BA 1
7
3
F
B8
5
9
A
D
H
E
G
C
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2
3
4
5
12 13 611
8
1210
6
1215
9
12
√5
A
B
1.2 m
2.6 m
A
B
1.7 cm
3 cm
3.5 cm
1.9 cm
BA
26 m
35 m
49 m
5.3 cm
5.3 cm
A
B
BA
2.1 km 1.8 km
0.5 km
AB
17.2 km 19.7 km
12.3 km
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x mm
11.4 mm
9.3
mm
x m
15.5 m
3.7 m 5 cm
3 cm
x cm
3 m
2 m
1.5 m
x m
6.2 km
8.2 km
6.7 km
x km 2.93 cm
4.04 cm5.31 cm
x cm
4
6
2
−4
−6
−2−4−6 −2−8
−8
4 62 8O
8
y
x
2 cm1 cm
r cm
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4.5 m
3 m
E
A B
C
GH
F
D
3.5 m
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Area = 6 m2
2 m
3 m
••
× 10002
÷ 10002
km2 m2
× 1002
÷ 1002
cm2
× 102
÷ 102
mm2
1 m
= 1
00 c
m
1 m = 100 cm
1 m2
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l w
l
h
b
yx h
b
h
b
a x
y
rθ
10 mm
10 mm
1 cm2
km m cm2 2 2
÷ 10002 ÷ 1002
1002 = 10 000 and 10002 = 1 000 000
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2 cm
3 cm
5 cm 5.8 m
3.3 m 1.06 km
Area = 11 mm2
l mm2.3 mm
Area = 0.5 m2
1.3 m
0.4 ma m
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∴
∴
( )
80°3 cm
5 m
◦ ◦ ◦
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5 cm
5.2 m
10.5 m
2.8 km
1.3 km
0.2 mm
0.1 mm
0.3 mm
7 m
15 cm10 cm
64 m 23 m 0.4 mm
0.25 mm
km23
km14
√3 km
0.82 m
1.37 m √6 cm√2 cm
√10 cm
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w cm
5.2 cm
l m
1.8 km
h km
1.3 m
2.8 m
a m
5.34 mm
h mm 18 m
Area = 80 m2x m
r cmd m
d m
6 cm
10 m
7 m
120°
100°2 m 150°
6 km3 mm
325°
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6.5 m
8.2 m
5 cm
4 m
7 m
10 m
1.7 m1.8 m
1.6 m
4.2 mm
28 km
18 km
26 km
0.25 m 0.3 m
0.7 m
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b
a
h
x
x
x
x
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�••
�•
h
r
h
r
r
2πr
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•
common face
3 cm
5 cm8 cm
5.3 m
1.7 m
3 cm
5 cm
8 cm
5 cm
3 cm
5.3 m2πr
1.7 m
1.7 m
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8 cm
10 cm
20 cm
6 cm
10 cm
4 cm
2 cm2 cm
1 cm3 cm
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3 cm
3 cm6 cm 2.7 mm
1.3 mm5.1 mm
2.1 cm
4.5 cm7 cm
8 m
17 m
15 m
20 m
12.8 m
9.2 m
5.2 m
6.4 m
1.2 cm
5 mm
3 mm
4 mm
7 mm
25 m10 m
8.66 m16 mm
5 mm
6 mm12 mm
1.2 cm
1.5 cm
0.8 cm0.76 cm
0.5 cm
2.3 m
4.8 m
2.1 m
1.4 m
1.9 m
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3 cm10 cm 2 m
1.6 m
6 m
4 m20 cm
40 cm
40°
7
3
2
1.4
1.8 3.2
8
7
10
8
10 33
26
1.85.2
2 cm
6 cm
4 cm
4 cm
10 m
8 m
10 cm
20 cm15 m
40 m20 m
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5 m2 m
x
a b
c
h
d rh
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r
r
y
x
r
h
θh
r
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••
s
rθ
sB
A
•••
•••
r
s rs
2πr
∴
s
apex
right coner
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••
apex
right square-based pyramid
4.5 m
2 m
25 mm
22 mm
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( )( )√( )
sh
r
5 cm
2 cm
14 m
5 m6 cm
10 cm
2 cm4 cm
2 cm
3 cm
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9 mm
12 mm
0.5 m
0.8 m
11 km
15 km
6 m
4 m
8 cm
5 cm0.3 m
0.2 m
10 m
2 m
1.1 cm
1.5 cm
26 mm
48 mm
6 m
4 m
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4 cm
5 cm
2 cm
3 cm
3 cm
4.33 cm
8 cm
5 cm
4.33 cm
10 m
3 m
8 m
10 mm
6 mm
9 mm 5 m
7 m2 m
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√
√
10 cm
10 cm
20 cm
8 cm
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11 cm
13 cm
7 cm5 cm 4 cm
8 cm
3 cm
13 cm
12 cm
x cm9 cm
40 cmx cm
8 cm
5 cmx cm
12 cm
4 mm 17.6 mm
5 cm
80°5 cm
4 cm
210°
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4 cm
6 cm7 cm
10 cm
5 cm
8 cm
3 cm
1.1 m
50 cm
12 cm
7.2 cm8 cm
6 cm
◦
120°7.5 cm
O
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10
63
10
63
10
63
••
•••
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× 10003
÷ 10003
km3 m3
× 1003
÷ 1003
cm3
× 103
÷ 103
mm3
•
× 1000
÷ 1000
ML kL
× 1000
÷ 1000
L
× 1000
÷ 1000
mL
••
w
h
l
h
xxr
h
4 m
3 m2 m
6 cm
2 cm
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3 cm8 cm
5 cm
6 cm2 cm
2 cm
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10 cm
2 m
12 mm
2 cm
4 cm5 cm
30 m10 m
35 m7 mm
3.5 mm10.6 mm
5 m
10 m1.5 cm
2 cm
12 m
14 m
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3 cm
2 cm
20 m
5 m
1.2 mm
2.4 mm
1.7 mm
2 km10 km
3 km
3.5 cm
8 cm
7 cm
7 m3 m
3 m
2 m
5 cm2 2 cm
0.12 m
0.38 m
21.2 cm
18.3 cm
24.5 cm
0.5 m
0.8 m
12 cm
12 cm
2 m
3 m
2 m
6 m
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8 cm
2 cm
5 cm
5 m
5 m
2.3 m
1.6 m
1.1 m 2.6 m
5 mm
8 mm11 mm 9 cm
3 cm
4 cm
5 cm
4 cm
6 cm
5
34
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h
r θ
10 m4 m
10 m
1 m
1 m
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••
•••
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xx
h h
r
h
r
1.4 m
1.2 m
1.3 m
29 mm
23 mm
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A = 5 m2
6 m4 cm
A = 2 cm2
7 mm
A = 25 mm2
2 cm
3 cm
18 m
15 m
13 m
12 km
5 km
5 km
2 cm
Perpendicularheight = 2 cm
4 cm
3 cm
Perpendicularheight = 4 cm
1.4 mm
0.6 mm
Perpendicularheight = 1.2 mm
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2.6 m
1.1 m
3.5 mm
1.6 mm
60 m
20 m
6 m
2 m
1.3 m
0.6 m
5 cm
20 cm
8 m
6 m
3 m
5 cm
6 cm
4 m
3 m6 m
2 m 4.8 mm
1.5 mm
5.2 mm
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3.96 m
2.8 m 7.1 m
10 m
2.6 m
6 cmice-cream
depth ofice-cream
3 cm
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r1
h1
r2
h2
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••
•
•
r
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r m
∴√
∴
5 cm 7 cm
√
√√
5 cm
2 cm
s cm
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( )
2 cm0.5 m
38 mm
18 cm0.92 km
1.36 m
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√ √
r cmr cm
r km
r m r cm r mm
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5 cm 4 cm 1 m
2 m4 m
5 m
1.3 cm
0.7 cm
4 mm
8 mm
1 m
5 m3 m
2 m
3 cm
2 cm
√2 cm
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3 m
1 m 13 m
19 m
2 cm
28 cm
20 cm
2 m
2 m
2 m
1 cm
hollow
15 cm10 cm
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xx
x
r
√
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•
•
•
•
85 86.0
86.585.5
87
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•
85.8 85.9 86.0 86.1 86.2
85.95 86.05
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4
26
6
3 9
6 m
2 m10 m
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1 m2 m
4 m2 m
1 m
10 cm
5 cm
6 cm
1 cmA
C
C
A
B
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10 cm
5 cm
2 cm
12 cm
6 cm
8 cm
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Up for a challenge?If you get stuck on a question,check out the 'Working withunfamiliar problems' poster
at the end of the bookto help you.
◦
45°
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Measurement
Pythagoras’ theorem
Volume
rh
h
r
Cone:
Sphere:
× 10003
÷ 10003
km3 m3
× 1003
÷ 1003
cm3
× 103
÷ 103
mm3
× 1000
÷ 1000
megalitres(ML)
kilolitres(kL)
millilitres(mL)
litres(L)
× 1000
÷ 1000
× 1000
÷ 1000
hw
lV = pr2hV = lwh
V = Ah,13
V = pr2h13
V = pr343
Perimeter is the distance aroundthe outside of a closed shape.
Circumference of a circleC = 2pr = pd
Perimeter of a sector
LengthUnits of length
× 1000
÷ 1000
km m
× 100
÷ 100
cm
× 10
÷ 10
mm
P = 2r + × 2prθ360
θ
r
cb
a
c2 = a2 + b2
Can occur in 3D shapes
AreaUnits of area
Formulas
× 10002
÷ 10002
km2 m2
× 1002
÷ 1002
cm2
× 102
÷ 102
mm2
sector rhombus parallelogram trapeziumkite
A = xyh 12A = xy1
2A = × pr2θ360
A = (a + b)A = bh 12
θ h
b
h
b
ax
yxy
where A is the area of the base.
r
circle
A = pr2
l
square
A = l 2
w
l
rectangle
A = lw
triangle
A = bh12
h
b
r
Usually ± 0.5 × the smallestunit of measuremente.g. 72 cm is 71.5 cm to 72.5 cm.
Limits of accuracy
Units of volume
rectangular prism
For pyramids and cones: (10A)
cylinder
Units of capacity
1 cm3 = 1 mL
For right and oblique prisms and cylindersV = Ah, where A is the area of the base h is the perpendicular height.
For prisms and pyramids draw thenet and add the areas of all the faces.
e.g.
Surface area
r
r
h h2pr
s
r
TSA = 4 × triangles + square base
TSA = 2prh + 2pr2
curved base surface ends
TSA = 4pr2
TSA = prs + pr2
curved + base
Cylinder
Cone
Sphere
For composite solids considerwhich surfaces are exposed.
+
(10A)
(10A)
(10A)
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x cm
7.1 cm10.4 cm
100°5 m
x km7 km
12
10
√ √ √√ √
7 m
12 m
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10 cm
9 cm
4 cm
8 m
8 m
9 m
7 m
4 m80°
6 cm
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D
H G
C
B
FE
A
2
47
5.2 m
6.2 m 12 m
18 m
13 m7 m
11 m
10 cm6.4 m
9.8 m
13 m
7 m4.2 m
8.1 m
w m
9.2 m
5.3 m
deck
6 m
5 m8 m 6 cm
10 cm
5 cm4 cm
8 cm
20 cm
10A13 cm
10 cm
10 cm
12 cm
10A
4 m
10A
12 cm
10 cm
13 cm
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10 cm
3 cm3 cm
7 m
4 m
5 m
2.5 mm 1.6 mm
0.9 mm
4 cm
5 cm
3.9 cm
5 cm
3 cm
2 cm 3.2 cm
1.8 cm
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39 cm
51 cm
36 cm
6 m
2 m 8 m
1 m
c m
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