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





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”


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


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!





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




19


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




24


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


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


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.




32


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


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




36


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





39


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




41


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


cm 63

3

cm 10,000 litres 10

cm 1000 litre 1

seconds 25 seconds 25.2539610,000




45


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




48


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


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


1y 2y 3y 4y 5y 6y 7y

h

26 252 1822 214 15 03

6

512 614152 2m 722102244152

area and volume lesson 1 lucan community college esker drive lucan, co dublin © 2010 ciarán duffy...

Documents

theorem of pythagoras

bc pythagoras theorem

babylonian mathematics

pictures pythagoras

number pythagoras theorem

blue square pythagoras

angles revision of area

bc bust of pythagoras