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Area and Volume Lesson 1 St. Joseph’s CBS Maths Department Fairview Dublin 3 © 2007 M Timmons December 3, 2007 Topics To be Covered Revision of Area & Perimeter

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A square is a polygon with four equal sides and angles Revision of Area and Perimeter Rectangle b l x x Square A rectangle is a quadrilateral where all four angles are right angles

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A parallelogram is a quadrilateral with two sets of parallel sides Revision of Area and Perimeter (cont.) Triangle base Parallelogram A triangle is a polygon with three sides and three angles hh b a

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A circle is a set of points the same distance ( r ) from a fixed point (centre) Revision of Area and Perimeter (cont.) CircleSector of a Circle Circle sector also known as pie piece, is a portion of a circle enclosed by two radii and an arc r

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The length of this sector is 1/6 the circumference of the circle 4/26/2015 7:53:39 PM Taking Ex 1 (i)Find the length of the perimeter of the sector oab.

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To get what fraction of a circle an angle is, put the angle over 360 and simplify 4/26/2015 7:53:39 PM Taking Ex 2 (i)Find the area of the sector aob.

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Area and Volume Lesson 2 St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons December 3, 2007 Topics To be Covered Theorem of Pythagoras

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Pythagoras (c. 580–500 BC) Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (situated in present day Iraq), from the days of the early Sumerians (3000 BC ) to the fall of Babylon in 539 BC Bust of Pythagoras in the Capitoline Museum Rome Theorem of Pythagoras In 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.

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The cuneiform script is one of the earliest known forms of written expression. Created by 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

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Some of the clay tablets contain mathematical lists and tables, others contain problems and worked solutions, 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.

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He and his followers believed “all 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: x 2 +y 2 =z 2 This is why the result bears his name.

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The sum of the area of the two green squares equals the area of the blue square Pythagoras Theorem (cont.)

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Originally built between c.3300-2900 BC according to Carbon-14 dates, it is more than 500 years older than 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 (21 st 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!

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Area and Volume Lesson 3 St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons December 3, 2007 Topics To be Covered Rectangular Solids Prisms

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A Rectangle Solid has a uniform cross section Rectangular Solid

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The area of the cross section in this example is the area of a triangle (half the base X perpendicular height) Prism: is a figure with a uniform cross section Volume = area of cross section X length

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Try the following example Ex 1 Find the volume of the following prism

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Try Questions from Text Book Ex 2 Find the volume of the following prism

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Area and Volume Lesson 4 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 19 Topics To be Covered Cylinder: Volume Curved Surface Area Total Surface Area

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A Cylinder has a circular top and bottom. The sides are vertical. The Cylinder

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A Cylinder is made up of a rectangular shape and two circles. The Surface Area of a Cylinder

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A can of beans is an example of a cylinder Ex 1

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Ex 2 Try this one yourself

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Area and Volume Lesson 5 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 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

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A football is an example of a sphere The Sphere radius =r

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A hemisphere is half a sphere The Hemisphere radius =r

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A sphere has the exact same appearance no matter what its viewing angle is Ex 1 Find (i) the volume (ii) the surface area of a sphere of radius 7 cm, take ∏=22/7 r = 7 cm

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Every point on the surfaces of a sphere is the same distance from its centre Q 1 Find (i) the volume (ii) the surface area of a sphere of radius 14 cm, take ∏=22/7 r = 14 cm

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Although the earth is not a perfect sphere the earth is divided into two hemispheres N and S by the equator Ex 2 r =7 cm Find (i) the volume (ii) the curved surface area of a hemisphere of radius 7 cm, take ∏=22/7

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The Northern Hemisphere contains most of the land and about 90 % of the human population. Q 2 r=12 cm Find (i) the volume (ii) the total surface area of a solid hemisphere of radius 12 cm, take ∏=3.14

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Because like other planets the earth is not a perfect sphere. The radius of the earth varies between 6356 km(Polar) and 6378 km (Equatorial), depending on where you are on the surface.

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Area and Volume Lesson 6 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 32 Topics To be Covered Volume of a Cone Curved Surface Area of a Cone Total Surface Area of a Cone

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A wizard’s hat is an example of a Cone Cone

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Try the following yourself: Cone Ex1 Find the Volume curved & the surface area of the following cone. Take ∏ =3.14

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Try Questions from Text Book Q 1 Find the Volume & the curved surface area of the following cone. Take ∏ =3.14

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Area and Volume Lesson 6 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 36 Topics To be Covered Compound 3D Shapes

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Compound Shapes 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. (i) Calculate the volume of the cone in terms of ∏. (iii) Find the total volume of the candle in terms of ∏. (ii) Find the height of the cylinder if the volume of the cylinder is twice that of the cone.

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Try the following yourself: 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. (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. Try Questions from Text Book

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Area and Volume Lesson 9 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 39 Topics To be Covered More Difficult type Questions: Involving ratios Where no values are given for r or h

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Ex 1: 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 Try Questions from Text Book

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Area and Volume Lesson 10 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 41 Topics To be Covered More Difficult type Questions: Liquids flowing through pipes

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Liquid flowing through a pipe 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. Remember the rate becomes the height

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Liquid flowing through a pipe. Example 1 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. Try the following yourself

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Liquid flowing through a pipe. Example 2 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. Try Questions from Text Book

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Area and Volume Lesson 11 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 45 Topics To be Covered Simpson’s Rule

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This 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: The strips are all of equal width h units

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EX 1 Find the area of the figure below, all figures in metres, give answer correct to 2 decimal places.

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Area and Volume Lesson 12 St. Joseph’s CBS Fairview Dublin 3 5.3 Maths 26 April 2015 48 Topics To be Covered Simpson’s Rule (Continued)

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EX 1 Find the area of the figure below, all figures in metres, give answer correct to 2 decimal places.

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EX 2 Find the area of the lake below, all figures in metres.

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 6 Geometry.

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