Area and Volume Lesson 1 Lucan Community College Esker Drive Lucan, Co Dublin © 2010 Ciarán Duffy...
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Transcript of Area and Volume Lesson 1 Lucan Community College Esker Drive Lucan, Co Dublin © 2010 Ciarán Duffy...
Area and VolumeArea and VolumeLesson 1Lesson 1
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
Topics To be Covered
Revision of Area & Perimeter
A square is a polygon with four A square is a polygon with four equal sides and anglesequal sides and angles
Revision of Area and PerimeterRectangle
b
l
blArea
bl 22Perimeter
x
x2Area x
x4Perimeter
Square
A rectangle is a quadrilateral where A rectangle is a quadrilateral where all four angles are right angles all four angles are right angles
A parallelogram is a quadrilateral A parallelogram is a quadrilateral with two sets of parallel sideswith two sets of parallel sides
Revision of Area and Perimeter (cont.)Triangle
base
heightbase2
1Area
hbArea
ba 22Perimeter
Parallelogram
A triangle is a polygon with three A triangle is a polygon with three sides and three angles sides and three angles
h h
b
a
A circle is a set of points the same A circle is a set of points the same distance (distance (rr) from a fixed point (centre)) from a fixed point (centre)
Revision of Area and Perimeter (cont.)Circle
2Area r
r2nceCircumfere
2
360sector of Area r
xaob
rx
ab 2360
arc ofLength
Sector of a Circle
Circle sector also known as pie piece, is a portion Circle sector also known as pie piece, is a portion of a circle enclosed by two radii and an arc of a circle enclosed by two radii and an arc
r
The length of this sector is 1/6 the The length of this sector is 1/6 the circumference of the circlecircumference of the circle
04/19/23 11:13 AM
7
22Taking Ex 1
(i) Find the length of the perimeter of the sector oab.
cm 67.14147
222
360
60
2360
arc ofLength
rx
ab
cm 67.4267.141414
ab arc length Total
oaob
To get what fraction of a circle an angle is, To get what fraction of a circle an angle is, put the angle over 360 and simplify put the angle over 360 and simplify
04/19/23 11:13 AM
7
22Taking
Ex 2
(i) Find the area of the sector aob.
2
2
cm 67.10214147
22
360
60360
sector of Area
rx
Area and VolumeArea and VolumeLesson 2Lesson 2
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
Topics To be Covered
Theorem of Pythagoras
Pythagoras (c. 580–500 BC)
Babylonian mathematics refers to any mathematics of the Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (situated in present day Iraq), from peoples of Mesopotamia (situated in present day Iraq), from the days of the early Sumerians (3000 BC ) to the fall of the days of the early Sumerians (3000 BC ) to the fall of Babylon in 539 BCBabylon in 539 BC
Bust of Pythagoras in the Bust of Pythagoras in the Capitoline Museum RomeCapitoline Museum Rome
Theorem of PythagorasIn a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides
This result was known long before this time.
Babylonian Mathematics records examples of this result.
The cuneiform script is one of the earliest The cuneiform script is one of the earliest known forms of written expression. Created known forms of written expression. Created by the Sumerians from ca. 3000 BCby the Sumerians from ca. 3000 BC
Pythagoras Theorem (cont.)
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include the Pythagorean theorem.
Our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s
Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun.
Babylonian Mathematics
Some of the clay tablets contain mathematical lists Some of the clay tablets contain mathematical lists and tables, others contain problems and worked and tables, others contain problems and worked solutions, others like above contain picturessolutions, others like above contain pictures
Pythagoras Theorem (cont.)The earliest tangible record of Pythagoras' Theorem comes from Babylonian tablets dating to around 1000 B.C. A number of tablets have been found with pictures which are in effect proofs of the Theorem in the special case where the sides of the right triangle are equal.
He and his followers believed “all He and his followers believed “all is number”is number”
Pythagoras Theorem (cont.)
Before Pythagoras, mathematicians did not understand that results, now called theorems, had to be proved.
So he was the first person to prove that:
x2+y2=z2
This is why the result bears his name.
The sum of the area of the two The sum of the area of the two green squares green squares equals the area of the blue equals the area of the blue squaresquare
Pythagoras Theorem (cont.)
169
25
222 534
Originally built between c.3300-2900 BC according to Originally built between c.3300-2900 BC according to Carbon-14 dates, it is more than 500 years older than Carbon-14 dates, it is more than 500 years older than the Great Pyramid of Giza in Egypt. the Great Pyramid of Giza in Egypt.
What was happening in Ireland around 3000 BC? (2500 years before Pythagoras)Newgrange: in County Meath, is one of the most famous prehistoric sites in the world. Newgrange is also one of the oldest surviving buildings in the world and was built in such a way that at dawn on the shortest day of the year, the winter solstice (21st December approx.), a narrow beam of sunlight for a very short time illuminates the floor of the chamber at the end of the long passageway. It is a World Heritage Site.
Perhaps the first calculator ever built!
Area and VolumeArea and VolumeLesson 3Lesson 3
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
Topics To be Covered
Rectangular Solids
Prisms
A Rectangle Solid has a uniform A Rectangle Solid has a uniform cross sectioncross section
Rectangular Solid
h
lb
hbl Volume
bhlhlb 222Area Surface
The area of the cross section in this The area of the cross section in this example is the area of a triangle example is the area of a triangle (half the base X perpendicular height)(half the base X perpendicular height)
Prism: is a figure with a uniform cross section
length
lengthsection cross of areaVolume
Volume = area of cross section X length
Try the following exampleTry the following example
Ex 1 Find the volume of the following prism
cm 10
3cm 24010682
1Volume
cm 8
cm 6
Try Questions from Text BookTry Questions from Text Book
Ex 2 Find the volume of the following prism
cm 12
3cm 25212762
1Volume
cm 6
cm 7
Area and VolumeArea and VolumeLesson 4Lesson 4
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
19
Topics To be Covered
Cylinder: Volume
Curved Surface Area
Total Surface Area
A Cylinder has a circular top and A Cylinder has a circular top and bottom. The sides are vertical.bottom. The sides are vertical.
The Cylinder
hr 2 Volume
A Cylinder is made up of a A Cylinder is made up of a rectangular shape and two circles.rectangular shape and two circles.
The Surface Area of a Cylinder
2r2rh2Area Surface Total
rh2Area Surface Curved
A can of beans is an example of a A can of beans is an example of a cylindercylinder
Ex 1
7
22 be to Take cm. 21 radius and cm 34height ith cylinder w
a of area surface curved the(ii) litres)in ( volume the(i) Find
litres 47.124 cm 47124
3421217
22 Volume (i)
3
2
hr
cm 4488
34217
2222 area Surface C. (ii)
2
rh
Ex 2 Try this one yourself
7
22 be to Take cm. 14 radius and 7cmheight ith cylinder w
a of area surface curved the(ii) litres)in ( volume the(i) Find
litres 4.312 cm 4312
714147
22 Volume (i)
3
2
hr
cm 616
7147
2222 area Surface C. (ii)
2
rh
Area and VolumeArea and VolumeLesson 5Lesson 5
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
24
Topics To be Covered
Volume of Sphere
Curved Surface Area of Sphere
Volume of Hemisphere
Curved Surface Area of Hemisphere
Total Surface Area of Solid Hemisphere
A football is an example of a A football is an example of a spheresphere
The Sphere
3
3
4 Volume r
radius =r
24 Area Surface r
A hemisphere is half a sphereA hemisphere is half a sphere
The Hemisphere
3
3
2 Volume r
22 Area surface Curved r
radius =r
2 topof Area r
A sphere has the exact same appearance no A sphere has the exact same appearance no matter what its viewing angle ismatter what its viewing angle is
Ex 1
3
3
cm 33.14377777
22
3
43
4 Volume i
r
24 Area surface Curved (ii) r
Find (i) the volume (ii) the surface area of a sphere of radius 7 cm, take ∏=22/7
r = 7 cm
2cm 616777
224
Every point on the surfaces of a Every point on the surfaces of a sphere is the same distance from its sphere is the same distance from its centrecentre
Q 1
3
3
cm 66.114981414147
22
3
43
4 Volume i
r
24 Area surface Curved (ii) r
Find (i) the volume (ii) the surface area of a sphere of radius 14 cm, take ∏=22/7
r = 14 cm
2cm 246414147
224
Although the earth is not a perfect sphereAlthough the earth is not a perfect spherethe earth is divided into two hemispheres N the earth is divided into two hemispheres N and S by the equatorand S by the equator
Ex 2
3
3
cm 67.7187777
22
3
2
3
2 Volume (i)
r
22 Area surface Curved (ii) r
r =7 cm
Find (i) the volume (ii) the curved surface area of a hemisphere of radius 7 cm, take ∏=22/7
2cm 308777
222
The Northern Hemisphere contains most of the The Northern Hemisphere contains most of the land and about 90 % of the human population.land and about 90 % of the human population.
Q 2
3
3
cm 28.361712121214.33
2
3
2 Volume (i)
r
23 Area surface Total (ii) r
r=12 cm
Find (i) the volume (ii) the total surface area of a solid hemisphere of radius 12 cm, take ∏=3.14
2cm 48.1356121214.33
Because like other planets the Because like other planets the earth is not a perfect sphere. The earth is not a perfect sphere. The radius of the earth varies between radius of the earth varies between 6356 km(Polar) and 6378 km 6356 km(Polar) and 6378 km (Equatorial), depending on where (Equatorial), depending on where you are on the surface. you are on the surface.
Area and VolumeArea and VolumeLesson 7Lesson 7
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
32
Topics To be Covered
Volume of a Cone
Curved Surface Area of a Cone
Total Surface Area of a Cone
A wizard’s hat is an example of a A wizard’s hat is an example of a ConeCone
Conehr 2
3
1 Volume
slant thecalled is
where
Area Surface C.
22
l
rhl
rl
2 Area Surface Total rrl
Try the following yourself:Try the following yourself:
Cone Ex1 Find the Volume curved & the surface area of the following cone. Take ∏ =3.14
3
22
cm 38.37
4314.33
1
3
1 Volume
hr
2
2222
cm 1.475314.3
Area Surface C.
534
rl
rhl
cm 4
cm 3
Try Questions from Text BookTry Questions from Text Book
Q1 Find the Volume & the curved surface area of the following cone. Take ∏ =3.14
3
22
cm 44.301
8614.33
1
3
1 Volume
hr
2
2222
cm 4.18810614.3
Area Surface C.
1068
rl
rhl
cm 8
cm 6
Area and VolumeArea and VolumeLesson 8Lesson 8
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
36
Topics To be Covered
Compound 3D Shapes
Compound Shapes
32
2 cm2
496
2
7
3
1
3
1 cone Volume
hr
42
492
4
49 cone. theof volume the twicecylinder Volume
hh
hhhr4
49
2
7 cylinder Volume
22
The diagram show the shape of a candle. It is made from a solid cylinder and a solid cone. The diameter at the base is 7 cm. The height of the cone 6 cm.
cm 7
cm 6
(i) Calculate the volume of the cone in terms of ∏.
(iii) Find the total volume of the candle in terms of ∏.
h
(ii) Find the height of the cylinder if the volume of the cylinder is twice that of the cone.
3cm 2
1474
4
49
2
49 Volume Total
Try the following yourself:
322 cm4323
1
3
1 cone Volume hr
4444
cone. theof volume thefour times cylinder Volume
hh
322 cm 42 cylinder Volume hhhr
The diagram show the shape of a candle. It is made from a solid cylinder and a solid cone. The diameter at the base is 4 cm. The height of the cone 3 cm.
cm 4
cm 3
(i) Calculate the volume of the cone in terms of ∏.
(ii) Find the height of the cylinder if the volume of the cylinder is four times that of the cone.
h
Try Questions from Text BookTry Questions from Text Book
Area and VolumeArea and VolumeLesson 9Lesson 9
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
39
Topics To be Covered
More Difficult type Questions:
•Involving ratios Where no values are given
for r or h
Ex 1:
hrhrhr 222 1232 cylinder Volume
hrhr 22 12:3
1 iscylinder :cone Volume
A cylinder has a radius that is twice the radius of a cone. The height of the cylinder is three times the height of the cone. Calculate the ratio of the volume of the cone: volume of the cylinder
h
h3
Try Questions from Text Book Try Questions from Text Book
r2
rhr 2
3
1 cone Volume
36:1 12:3
1 iscylinder :cone Volume
Area and VolumeArea and VolumeLesson 10Lesson 10
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
41
Topics To be Covered
More Difficult type Questions:
•Liquids flowing through pipes
Liquid flowing through a pipe
322 cm 6373 cylinder of Volume hr
When liquid flows through a cylindrical pipe of radius 3 cm, at the rate of 7 cm/sec. The volume that passes through the pipe in 1 second is the same as the volume of a cylinder with radius 3 cm and height 7 cm. Here the rate becomes the height.
cm 7
Remember the rate becomes the Remember the rate becomes the heightheight
cm 3
Liquid flowing through a pipe. Example 1
32
2
cm 88727
22
cylinder of Volume
hr
Liquid flows through a cylindrical pipe of internal diameter 4 cm, at the rate of 7 cm/sec. How long to the nearest minute, will it take to fill a 20 litre bucket. Take ∏=22/7.
cm 7
Try the following yourselfTry the following yourself
cm 43
3
cm 20,000 litres 20
cm 1000 litre 1
minutes 4 minutes 3.79 seconds 227.278820,000
Liquid flowing through a pipe. Example 2
32
2
cm 3961437
22
cylinder of Volume
hr
Liquid flows through a cylindrical pipe of internal diameter 6 cm, at the rate of 14 cm/sec. How long to the nearest second, will it take to fill a 10 litre bucket. Take ∏=22/7.
cm 14
Try Questions from Text BookTry Questions from Text Book
cm 63
3
cm 10,000 litres 10
cm 1000 litre 1
seconds 25 seconds 25.2539610,000
Area and VolumeArea and VolumeLesson 11Lesson 11
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
45
Topics To be Covered
Simpson’s Rule
Simpson’s RuleThis rule is used to calculate the area of shapes with irregular boundaries. It involves dividing the shape into strips of equal
width h units. Simpson’s Rule is one method of finding the total area of these strips. The vertical lines are
called the offsets or ordinates:
.........,, 321 yyy
1y 2y 3y 4y 5y 6y 7y 8y 9y
The strips are all of
equal width h units
h h h h h h h h ordinatelast theis
ordinatefirst theis
9
1
y
y
Odds2Evens4Last First 3
Rule sSimpson' h
753864291 2 4 3
Ex. above In the yyyyyyyyyh
EX 1Find the area of the figure below, all figures in metres, give answer correct to 2 decimal places.
6 10 11 10 10 9 8
4 Odds2Evens4Last First 3
Rule sSimpson' h
5364271 2 4 3
Ex. above In the yyyyyyyh
1y 2y 3y 4y 5y 6y 7y
h
10 112 910 104 8 63
4
212 294143
4 2m 33.2294211614
3
4
Area and VolumeArea and VolumeLesson 12Lesson 12
Lucan Community CollegeEsker Drive
Lucan, Co Dublin © 2010 Ciarán Duffy
48
Topics To be Covered
Simpson’s Rule (Continued)
EX 1Find the area of the figure below, all figures in metres, give answer correct to 2 decimal places.
8 12 13 12 12 11 10
5 Odds2Evens4Last First 3
Rule sSimpson' h
5364271 2 4 3
Ex. above In the yyyyyyyh
1y 2y 3y 4y 5y 6y 7y
h
12 132 1112 124 10 83
5
252 354183
5 2m 67.3465014018
3
5
Odds2Evens4Last First 3
Rule sSimpson'
h
EX 2Find the area of the lake below, all figures in metres.
0 21 25 22 26 1815
6
5364271 2 4 3
Ex. above In the yyyyyyyh
1y 2y 3y 4y 5y 6y 7y
h
26 252 1822 214 15 03
6
512 614152 2m 722102244152