area & volume learning outcomes find the area of square, rectangles, triangles, parallelograms,...
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Area & VolumeLearning 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.
Find the surface areas and volumes of prisms.
Find the surface area and volume of the cylinder to include inverse calculations
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 & Volume Area of Shapes
Triangles
Area = ½ base × perp. heightA = ½ b × h
5cm
6cm
60°8cm
12cm
A
C B
b
a
c A = ½ ab sin C
Area = ½ product of 2 sides × sine of angle between them
Area & Volume Area of Shapes
Trapezium
One pair of parallel sides
Total area = bh + ½.ah - ½.bh = ½.bh + ½.ah = h/2(a+b)
Area 1 → b × h
h
b
a
21
a - b
Area 2 → ½ (a – b) × h
Area of a trapezium = ½ (a +b) × h = h/2(a+b)
Area & Volume Area of Shapes
Parallelogram
Area = a × b = length × perp. height
a
b
Kite
Area = ½ ab = ½ × length × width
b
a
Area & Volume Area of Shapes
Find the area of the following shapes
3cm
5cm
10cm
7cm
1. 8cm
6cm 12cm
2.
Area & Volume Area and Perimeter
Circles
O
C
BA
E
D
AB – diameter of circle
OC – radius of circle
DE – chord of circle
Area of a circle A = π r2
Perimeter of a circleCircumference C = π d C = 2 π r
Area & Volume Area and Perimeter
Find the area and perimeter of
12cm
12cm
8cm
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 CircleArea and Perimeter
Consider a sector of a circle which subtends an angle of 60° at the centre.
Find (i) The area of this sector(ii) The arc length AB(iii) The perimeter of the shape OAB
60°
r =5cm
O
BA
Area & Volume Sectors of a CircleArea and Perimeter
In general
θ° r
Area = θ × π r2
360
Arc length = θ × 2 π r 360
Find(i) The area of the sector OAB (ii) The area of the triangle
OAB
(i) The area of the minor segment (shaded)
Area & Volume Sectors, Segments, Chords and Area
O
BA
7150°
OA = radiusAB = chord
The chord AB subtends an angle of 150° at the centre of the circle. The radius of the circle is 7cm.
Area & Volume Trigonometry
sin x = opposite . hypotenuse
cos x = adjacent . hypotenuse
tan x = opposite adjacent
Remember
SOH CAH TOA
8 x
30°
1.
O
A
H 15 x
40°2.
A
O
H
3x
40°
3.
O
A
H
Area & Volume Volume of Prism
Volume of a prism = cross sectional area × length
Cylinder
V = π r2 h
r h V
3
3.2
4
5 128
125
Complete the table below (cylinders)
a)
b)
c)
lh
a
b
Area & Volume Volume of Prism
Volume of a prism = cross sectional area × length
Trapezoidal
V = ½ (a + b)h × l
104 5
8
Cross-sectional area =
Volume V =
Example
Area & Volume Volume of Prism
ConePyramid Sphere
h
ba
r
hl
r
a, b = dimensions of base h = perpendicular height
r = radiush = perpendicular heightl = slant length V = 4/3 π r3
V = 1/3 π r2 hV = 1/3 × base area × height
r = radius
Find the volume of(i) Pyramid (ii) Cone (iii) Sphere a = 3, b = 4, h = 2 r =2, h = 4.5 r = 1.5
Area & Volume Volume of Prism
Find r.
r
6
V = 300cm3
Find r.
r
V = 250cm3
10
h
V = 500cm3
Find h.
h
510
Find h.
V = 210cm3
Area & VolumeSurface Area of a
Cone, Sphere & Cylinder
r
hl
Cone
SA = π rl
h
r
l l2 = r2 + h2
Remember…
r
h
cylinder
SA = 2 π rh + 2 π r2
SA = Curved surface area + Top + Bottom
h
2 π rCurved area = = 2 π rh
r
Sphere
SA =4 π r2
Area & VolumeSurface Area of a
Cone, Sphere & Cylinder
Calculate the Surface Area of
3
5
b)
6
l6
a)
c)
5.2
r = 1.8
5
d)
e)14.8
4
Area & Volume SOH CAH TOA
Use Trig to find angle marked x in the following triangles
5cm3cm
x
3cm
5cm
x
3cm
5cm
x
10.410.4
14.6
x
Area & Volume Using Trig in Area Calculations
O
BA
10.4
14.6
10.4
Find(i) The angle AOB(ii) The area of sector AOB(iii) Area of triangle AOB(iv) The shaded area
Area & Volume Dimensions
Distinguishing between length, area & volume
r
h
Formulae can be used to calculate length e.g.
VolumeV = π r2 h
L2 × L2 = L3
(Volume)
b
a
PerimeterP = 2a + 2b
L + L = L
(Length)
b
a
Area A = a × b
L × L = L2
(Area)
Area & Volume Dimensions
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
Area & Volume Additional Notes
Area & Volume
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.
Learning Outcomes:At the end of the topic I will be able to
Can Revise Do Further
Find the surface area and volume of the cylinder to include inverse calculations
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.