Transcript c) Areas and Volumes - Student - school

Area & Volume
Learning Outcomes

Find the area of square, rectangles, triangles, parallelograms, rhombuses, kites,
trapezia and shapes which are composites of these.

Find the circumference and area of a circle, to include inverse calculations.

Find the area and circumference of a sector and segment of a circle.
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Find the surface areas and volumes of prisms.

Find the surface area and volume of the cylinder to include inverse calculations
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Find the surface area and volumes of the cone and sphere to include inverse
calculations.
Find the surface areas and volumes of compound shapes most of which are
everyday objects.
Distinguish between formulae for perimeter, area and volume by considering
dimensions.
Area of Shapes
Area & Volume
Triangles
Area = ½ base × perp. height
A=½b×h
5cm
6cm
A
A = ½ ab sin C
c
b
C
a
B
Area = ½ product of 2 sides
× sine of angle between them
8cm
60°
12cm
Area of Shapes
Area & Volume
Trapezium
One pair of parallel sides
b
2
a-b
Area 1 → b × h
Area 2 → ½ (a – b) × h
1
h
a
Total area = bh + ½.ah - ½.bh
= ½.bh + ½.ah
= h/2(a+b)
Area of a trapezium = ½ (a +b) × h
= h/2(a+b)
Area of Shapes
Area & Volume
Parallelogram
Kite
b
a
b
a
Area = a × b
= length × perp. height
Area = ½ ab
= ½ × length × width
Area of Shapes
Area & Volume
Find the area of the following shapes
1.
5cm
3cm
2.
7cm
10cm
8cm
6cm
12cm
Area and Perimeter
Area & Volume
Circles
C
A
O
AB – diameter of circle
B
E
OC – radius of circle
DE – chord of circle
D
Area of a circle A = π r2
Perimeter of a circle
Circumference C = π d
C=2πr
Area & Volume
Area and Perimeter
Find the area and perimeter of
12cm
8cm
12cm
Area & Volume
Area and Perimeter
1. The area of a circle is 51.6cm2. Calculate its radius.
2. The circumference of a circle is 50cm. Calculate its area.
Area & Volume
Area and Perimeter
3. The radius of a car wheel is 53cm. How many revolutions does
the wheel make in travelling 50m?
?
Area & Volume
Area and Perimeter
4. Find the shaded area.
8cm 10cm
Area & Volume
Sectors of a Circle
Area and Perimeter
O
Consider a sector of a circle which
subtends an angle of 60° at the centre.
60°
r =5cm
Find
(i) The area of this sector
(ii) The arc length AB
(iii) The perimeter of the shape OAB
A
B
Area & Volume
Sectors of a Circle
Area and Perimeter
In general
θ°
Area = θ × π r2
360
r
Arc length = θ × 2 π r
360
Sectors, Segments,
Chords and Area
Area & Volume
OA = radius
AB = chord
O
150°
A
7
B
The chord AB subtends an angle of
150° at the centre of the circle. The
radius of the circle is 7cm.
Trigonometry
Area & Volume
sin x = opposite .
hypotenuse
Remember
SOH CAH TOA
cos x = adjacent .
hypotenuse
tan x = opposite
adjacent
2.
1.
H
8
x O
H
3.
40°
15
x A
30°
H
x
3 O
40°
A
O
A
Volume of Prism
Area & Volume
Volume of a prism = cross sectional area × length
Cylinder
V = π r2 h
Complete the table below (cylinders)
r
a)
3
3.2
V
4
5
b)
c)
h
128
125
Volume of Prism
Area & Volume
Volume of a prism = cross sectional area × length
Trapezoidal
V = ½ (a + b)h × l
a
h
l
b
Example
Cross-sectional area =
Volume V =
4
5
10
8
Area & Volume
Volume of Prism
Cone
Pyramid
h
Sphere
l
h
b
a
a, b = dimensions of base
h = perpendicular height
V=
1/
3
× base area × height
r
r
r = radius
h = perpendicular height
l = slant length
r = radius
V = 4/3 π r3
V = 1/3 π r2 h
Find the volume of
(i) Pyramid
a = 3, b = 4, h = 2
(ii) Cone
r =2, h = 4.5
(iii) Sphere
r = 1.5
Area & Volume
Volume of Prism
Find r.
Find h.
r
h
10
V = 500cm3
V = 250cm3
Find h.
Find r.
h
6
r
V = 300cm3
10
5
V = 210cm3
Surface Area of a
Cone, Sphere & Cylinder
Area & Volume
Sphere
Cone
Remember…
l
l
h
h
r
l2 = r2 + h2
r
r
SA = π rl
SA =4 π r2
cylinder
r
SA = Curved surface area + Top + Bottom
h
2πr
Curved area =
= 2 π rh
h
SA = 2 π rh + 2 π r2
Area & Volume
Surface Area of a
Cone, Sphere & Cylinder
Calculate the Surface Area of
a)
d)
5
5
3
e)
b)
l
6
4
6
c)
5.2
r = 1.8
14.8
Area & Volume
SOH CAH TOA
Use Trig to find angle marked x in the following triangles
3cm
x
5cm
3cm
5cm
x
3cm
10.4
x
10.4
5cm
x
14.6
Using Trig in Area
Calculations
Area & Volume
O
10.4
10.4
14.6
A
B
Find
(i) The angle AOB
(ii) The area of sector AOB
(iii) Area of triangle AOB
(iv) The shaded area
Dimensions
Area & Volume
Distinguishing between length, area & volume
Formulae can be used to calculate length e.g.
b
Volume
V = π r2 h
r
h
L2 × L2 = L3
a
L+L=L
(Volume)
(Length)
b
a
Perimeter
P = 2a + 2b
Area
A=a×b
L × L = L2
(Area)
Area & Volume
Further Examples
a) 4 l + 4 b + 4 h
b) (l + b)2h2
c) (2π)rh
d) π(rh2 – 1/3 h3)
e) πr
(r2 + h2)
f) 1/3b2h
Dimensions
Area & Volume
Additional Notes
Area & Volume
Learning Outcomes:
At the end of the topic I will be able to
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


Find the area of square, rectangles, triangles,
parallelograms, rhombuses, kites, trapezia and
shapes which are composites of these.
Find the circumference and area of a circle, to
include inverse calculations.
Find the area and circumference of a sector and
segment of a circle.
Find the surface areas and volumes of prisms.
Can
Do
Revise
Further
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Find the surface area and volumes of the cone and
sphere to include inverse calculations.
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Find the surface areas and volumes of compound
shapes most of which are everyday objects.


Distinguish between formulae for perimeter, area
and volume by considering dimensions.
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
Find the surface area and volume of the cylinder to
include inverse calculations