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Ch 4 . Measuring Prisms and Cylinders . Area of a Rectangle . To find the area of a rectangle, multiply its length by its width. A = l x w. How do you find the area. So, the area is 8.36 cm 2 . Area of a Triangle .

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Ch 4

Measuring Prisms and Cylinders


Area of a rectangle
Area of a Rectangle

  • To find the area of a rectangle, multiply its length by its width.

  • A = l x w


How do you find the area
How do you find the area

  • So, the area is 8.36 cm2.


Area of a triangle
Area of a Triangle

  • To find the area of a triangle, multiply its base by its height, then divide by 2.

  • Remember the height of a triangle is perpendicular to its base.

  • The formula for the area of a triangle can be written:


How do you find the area1

.

How do you find the area

Substitute b = 12 and h = 3.

So, the area is 18 m2.


Watch brainpop area of polygons
Watch Brainpop: Area of Polygons

  • http://www.brainpop.com/math/geometryandmeasurement/areaofpolygons/preview.weml


Area of a circle
Area of a Circle

  • To find the area of a circle, use the formula:

    • A = r 2

    • where r represents the radius of the circle

  • Recall that  is a non-terminating and a non-repeating decimal number. So, any calculations involving  are approximate.

  • You need to use the  function on your calculator to be more accurate – 3.14 is not accurate enough.


  • How do you find the area2
    How do you find the Area

    Use A = r 2.

    Substitute r = 6 ÷ 2 = 3.

    So, the area is about 28 mm2.


    Circumference of a circle
    Circumference of a Circle

    • The perimeter of a circle is named the circumference.The circumference is given by: C = d or C = 2r (Recall: d = 2r)


    Find the circumference of the circle
    Find the circumference of the circle.

    The circumference of the circle is about 22 cm.


    Find the radius of the circle
    Find the radius of the circle.

    • The circumference of a circle is 12.57 cm

    • To find the radius of the circle, divide the circumference by 2.

    The radius of the circle is about 2 cm.


    Watch brainpop circles
    Watch Brainpop: Circles

    • http://www.brainpop.com/math/geometryandmeasurement/circles/preview.weml

    • http://www.youtube.com/watch?v=lWDha0wqbcI


    Try it
    Try It

    • Workbook pg 74 – 75

    • Puzzle Package



    Watch brainpop review
    Watch Brainpop: Review

    • http://www.brainpop.com/math/numbersandoperations/pi/preview.weml

    • http://www.brainpop.com/math/geometryandmeasurement/polygons/preview.weml

    • http://www.brainpop.com/math/geometryandmeasurement/polyhedrons/preview.weml

    • http://www.brainpop.com/math/geometryandmeasurement/typesoftriangles/preview.weml


    4 1 exploring nets

    4.1 Exploring Nets

    http://www.youtube.com/watch?v=y0IDttNW1Wo&feature=related


    A prism
    A Prism

    • A prism has 2 congruent bases and is named for its bases.

    • When all its faces, other than the bases, are rectangles and they are perpendicular to the bases, the prism is a right prism.

    • A regular prism has regular polygons as bases.



    Pyramid
    Pyramid

    • A regular pyramid has a regular polygon as its base. Its other faces are triangles. They are named after its base.

    Regular OctagonalPyramid

    Regular Pentagonal Pyramid

    Regular Square Pyramid


    Describe the faces of
    Describe the faces of:

    • ACube

      • 6 congruent squares

    • A Right Square Pyramid

      • 1 square, 4 congruent isosceles triangles

    • A Right Pentagonal Prism

      • 2 regular pentagons, 5 congruent rectangles


    Nets

    • A net is a diagram that can be folded to make an object. A net shows all the faces of an object

    • http://www.senteacher.org/wk/3dshape.php


    Nets

    • A net is a two-dimensional shape that, when folded, encloses a three-dimensional object. http://www.youtube.com/watch?v=9KXuaT18Jyw&feature=related (quick)

    • The same 3-D object can be created by folding different nets.

    • You can draw a net for an object by visualizing what it would look like if you cut along the edges and flattened it out.

      • http://www.youtube.com/watch?v=rMNa9wICWbo&feature=related



    Turn to p 178 of textbook
    Turn to p. 178 of Textbook

    This diagram is not a rectangular prism!

    Here is the net being put together


    P 178 of textbook example 1a
    p. 178 of TextbookExample #1a

    2 congruent regular pentagons

    5 congruent rectangles


    P 178 of textbook example 1b
    p. 178 of TextbookExample #1b

    1 square

    4 congruent isosceles triangles


    P 178 of textbook example 1c
    p. 178 of TextbookExample #1c

    This net has 2 congruent equilateral triangles and 3 congruent rectangles.

    The diagram is a net of a right triangular prism. It has equilateral triangular bases.


    P 178 of textbook example 1 d
    p. 178 of TextbookExample #1d

    This is not a net. The two triangular faces will overlap when folded, and the opposite face is missing.

    Move one triangular face from the top right to the top left. It will now make a net of an octagonal pyramid.


    P 179 of textbook example 2
    p. 179 of TextbookExample #2



    What is surface area
    What is Surface Area?

    • Surface area is the number of square units needed to cover a 3D object

    • It is the sum of the areas of allthe faces of an object


    Finding sa of a rectangular prism
    Finding SA of a Rectangular Prism

    • The SA of a rectangular prism is the sum of the areas of its rectangular faces. To determine the surface area of a rectangular prism, identify each side with a letter.

      • Rectangle A has an area of

        • A = l x w

        • A = 4 x 5

        • A = 20

      • Rectangle B has an area of

        • A = l x w

        • A = 7 x 5

        • A = 35

      • Rectangle C has an area of

        • A = l x w

        • A = 7 x 4

        • A = 28

    C

    B

    A


    Finding sa of a rectangular prism1
    Finding SA of a Rectangular Prism

    • To calculate surface area we will need 2 of each side and add them together

      • SA = 2(A) + 2(B) + 2(C)

      • SA = 2(l x w) + 2(l x w) + 2(l x w)

      • SA= 2(4 x 5) + 2(7 x 5) + 2(4 x 7)

      • SA = 2(20) + 2(35) + 2(28)

      • SA = 40 + 70 + 56

      • SA = 166 in2

    C

    B

    A


    Finding sa of a rectangular prism2
    Finding SA of a Rectangular Prism

    • Another Example

      • SA = 2(A) + 2(B) + 2(C)

      • SA = 2(l x w) + 2(l x w) + 2(l x w)

      • SA= 2(8 x 10) + 2(7 x 8) + 2(10 x 7)

      • SA = 2(80) + 2(56) + 2(70)

      • SA = 160 + 112 + 140

      • SA = 412 units2

    C

    B

    A


    Finding sa of a rectangular prism3
    Finding SA of a Rectangular Prism

    • You Try

      • SA = 2(A) + 2(B) + 2(C)

      • SA = 2 (l x w) + 2(l x w) + 2 (l x w)

      • SA= 2(15 x 6) + 2(10 x 6) + 2(10 x 15)

      • SA = 2(90) + 2(60) + 2(150)

      • SA = 180 + 120 + 300

      • SA = 600 cm2

    C

    B

    A


    Some video reminders
    Some Video Reminders

    • http://www.youtube.com/watch?v=oR1ukNC1pvA

    • http://www.youtube.com/watch?v=agIV623B3nc&feature=related


    Practice
    Practice

    • Pg p.186

    • #4, 6,7,10,12,13,15



    Surface area of a right triangular prism
    Surface area of a Right Triangular Prism

    • To calculate the surface area of right triangular prism, draw out the net and calculate the surface area of each face and add them together.


    Surface area of a right triangular prism1
    Surface area of a Right Triangular Prism

    • Draw and label the net

    20cm

    16cm

    D

    16cm

    20cm

    12cm

    10cm

    B

    10cm

    C

    10cm

    A

    D

    16cm

    20cm


    Surface area of a right triangular prism2
    Surface area of a Right Triangular Prism

    • SA = A area + B area + C area + 2 D area

    • SA = A(l x w) +B(l x w) + C(l x w) + 2 * D[(b x h) ∕ 2)]

    • SA = A(16 x 10) + B(10 x 12) + C(20 x 10) + 2 * D[(16 x 12) ∕ 2]

    • SA = 160 + 120 + 200 + (2 * 96)

    • SA = 160 + 120 + 200 + 192

    • SA = 672 cm2

    20cm

    16cm

    D

    16cm

    20cm

    12cm

    10cm

    B

    10cm

    C

    10cm

    A

    D

    16cm

    20cm


    Surface area of a right triangular prism3
    Surface area of a Right Triangular Prism

    • Draw and label the net

    10cm

    6cm

    D

    6cm

    10cm

    8cm

    15cm

    B

    15cm

    C

    15cm

    A

    D

    6cm

    10cm


    Surface area of a right triangular prism4
    Surface area of a Right Triangular Prism

    • SA = A area + B area + C area + 2 D area

    • SA = A(l x w) + B(l x w) + C(l x w) + 2 * D[(b x h) ∕ 2)]

    • SA = A(15 x 6) + B(15 x 8) + C(15 x 10) + 2 * D[(6 x 8) ∕ 2]

    • SA = 90 + 120 + 150 + (2 * 18)

    • SA = 90 + 120 + 150 + 36

    • SA = 396 cm2

    10cm

    6cm

    D

    6cm

    10cm

    8cm

    15cm

    B

    15cm

    C

    15cm

    A

    D

    6cm

    10cm


    Surface area of a right triangular prism5
    Surface area of a Right Triangular Prism

    • Draw and label the net

    2.7m

    1.4m

    D

    1.4m

    2.7m

    2.3m

    0.7m

    B

    0.7m

    C

    0.7m

    A

    D

    1.4m

    2.7m


    Surface area of a right triangular prism6
    Surface area of a Right Triangular Prism

    • SA = A area + B area + C area + 2 D area

    • SA = (la x wa) + (lb x wb) + (lc x wc) + 2[(ld x wd) ∕ 2)]

    • SA = (1.4 x 0.7) + (0.7 x 2.3) + (2.7 x 0.7) + 2 [(1.4 x 2.7) ∕ 2]

    • SA = 0.98 + 1.61 + 1.89 + 2(1.89)

    • SA = 8.26m2

    2.7m

    1.4m

    D

    1.4m

    2.7m

    2.3m

    0.7m

    B

    0.7m

    C

    0.7m

    A

    D

    1.4m

    2.7m


    Surface area of a right triangular prism7
    Surface area of a Right Triangular Prism

    • Draw and label the net

    ? m

    3m

    D

    3m

    ?m

    8cm

    7 m

    B

    7 m

    7 m

    C

    7 m

    A

    D

    3cm

    ?m


    Surface area of a right triangular prism8
    Surface area of a Right Triangular Prism

    • Before you can solve, you need to find the missing side using Pythagorean Theorem!

    ? m

    3m

    D

    3m

    ?m

    8cm

    7 m

    B

    7 m

    C

    7 m

    A

    D

    3cm

    ?m


    Surface area of a right triangular prism9
    Surface area of a Right Triangular Prism

    • Pythagorean Theorem

    • a2 + b2 = h2

    • 32 + 82 = h2

    • 9 + 64 = h2

    • 73 = h2

    • √73 = √ h2

    • 8.54 = h

    8.54m

    3m

    8.54m

    3m

    8m

    7 m

    7 m

    7 m

    3m

    ?m


    Surface area of a right triangular prism10
    Surface area of a Right Triangular Prism

    • SA = A area + B area + C area + 2 D area

    • SA = A(l x w) + B(l x w) + C(l x w) + 2 * D[(l x w) ∕ 2)]

    • SA = A(3 x 7) + B(7 x 8) + C(8.54 x 7) + 2 * D[(3x 8) ∕ 2]

    • SA = 21 + 56 + 59.78 + (2 * 12)

    • SA = 21 + 56 + 59.78 + 24

    • SA = 160.78 m2

    8.54m

    3m

    D

    8.54m

    3m

    8m

    7 m

    B

    C

    7 m

    7 m

    A

    D

    3m

    ?m


    Some review videos
    Some Review Videos

    • http://www.youtube.com/watch?v=9pQknZ9fRTA

    • http://www.youtube.com/watch?v=mL7E_NBhyIw&feature=related


    Practice1
    Practice

    • P. 191



    Videos
    Videos

    • VIDEO Volume of a right rectangle

    • http://www.youtube.com/watch?v=HYyBP4K65TI


    The formula
    The Formula

    • V = Ah

    • Volume equals the area of the base of the rectangle times its height – this applies to all right cylinders or prisms

    • http://www.youtube.com/watch?v=QR22Zvpj3L4&feature=relmfu


    Example 1
    Example 1

    • V = Ah

    • V = (l x w) x h

    • V = (13.5 x 5) x 18.5

    • V = 67.5 x 18.5 (not really needed)

    • V = 1248.75 cm3


    Example 2
    Example 2

    Jack’s family opened a full carton of frozen yogurt for dessert. After they ate, there was ¾ left. Jack wants to know what volume of frozen yogurt they ate. He measured the carton to be to have a length of 12cm, a width or 9 cm and a height of 18cm.

    • V = Ah

    • V = (l x w) x h

    • V = (12 x 9) x 18

    • V = 1944 cm3 (Volume of entire container – we want ¾ of the container )

    • V = 1944 x ¾

    • V = 1458 cm3


    Workbook
    Workbook

    • Read “Quick Review” on pg. 85

    • Complete pg 85-86



    Video clip
    Video Clip

    • Relationship of a right triangular prism with right rectangular prism

    • http://www.youtube.com/watch?v=NqlyyVrHnp8


    Formula
    Formula

    • Volume is equal to the area of the base of the triangle times the length

      • V = Al


    Example 11
    Example 1

    • V = Al

  • V = (b * h /2) x l

  • V = (3 x 8 /2) x 7

  • V = (24 /2) x 7

  • V = 12 x 7

  • V = 84 m3


  • Example 21
    Example 2

    • A glass vase in the shape of a right triangular prism is filled with colouredsand as a decoration. What is the volume of the vase. It measured 15cm by 8cm by 80cm

    • V = Al

    • V = (b * h /2) x l

    • V = (15 x 8 /2) x 80

    • V = 60 x 80

    • V = 4800 cm3


    Watch brainpop volume of prisms
    Watch Brainpop: Volume of Prisms

    • http://www.brainpop.com/math/geometryandmeasurement/volumeofprisms/preview.weml


    Workbook1
    Workbook

    • Complete pg. 87 - 89



    Formula1
    Formula

    • http://www.youtube.com/watch?v=GpHhvuRbB-8


    • Imagine a can. If you unwrap the label off the can, the lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.


    Notes
    Notes lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • To find the surface area of a cylinder – sketch the net

    • Remember that the width of the rectangle will be equal to the circumference of the circle (2πr)

    • SA = 2(area of a circle) + area of the rectangle


    Example 12
    Example 1 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    SA = 2πr2 + (2πr)(h)

    SA = 2π(3.1)2 + (2π(3.1))(12)

    SA = 2π(9.61)+ (2π(3.1))(12)

    SA = 120.762822 + 230.734493

    SA = 354.50inches2


    Example 22
    Example 2 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    SA = 2πr2 + (2πr)(h)

    SA = 2π(9)2 + (2π(9))(11)

    SA = 2π(81)+ (2π(9))(11)

    SA = 508.938010 + 622.035345

    SA = 1130.97m2

    11m

    9m


    Example 3
    Example 3 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • Calculate the outside surface area of the cylinder. The cylinder is open on one end.

      SA = πr2 + (2πr)(h)

      SA = π(3)2 + (2π(3))(11)

      SA = π(9)+ (2π(3))(11)

      SA = 28.274333 + 207.345115

      SA = 235.62m2

    8m

    3m


    Review video
    Review Video lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • http://www.youtube.com/watch?v=5GODwuQxYNo&feature=related


    Workbook2
    Workbook lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • Complete pg 90 - 92


    4 8 volume of a right cylinder

    4.8 Volume of a Right Cylinder lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.


    The formula1
    The Formula lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • Cylinders follow the same properties as prisms for volume

    • V = area of the base x height

    • V = πr2 x h


    Example 13
    Example 1 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    V = πr2 x h

    V = π(22)x 6

    V = π(4) x 6

    V = 12.6 x 6

    V = 75.6 m3

    6m

    2m


    Example 23
    Example 2 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • Martha has a choice of two different popcorn containers at a movie. Both containers are the same price. Which container should Martha buy is she wants more popcorn for her money.

    • Container 1: has a diameter of 20cm and height of 40cm.

    • Container 2 has a diameter of 30cm and a height of 20cm.


    Container 1 lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    Container 2

    Radius is half the diameter so r = 15cm

    V = πr2 x h

    V = π(152)x 20

    V = π(225) x 20

    V = 706.9 x 20

    V = 14138 cm3

    • Radius is half the diameter so r = 10cm

      V = πr2 x h

      V = π(102)x 40

      V = π(100) x 40

      V = 314.2 x 40

      V = 12568 cm3


    Watch brainpop volume of cylinders
    Watch lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.Brainpop: Volume of Cylinders

    • http://www.brainpop.com/math/geometryandmeasurement/volumeofcylinders/preview.weml


    Workbook3
    Workbook lateral surface, or label, would be a rectangle. The height of the label is the same height as the can. The length of the label is the diameter of the base of the can because the label wraps all the way around the can.

    • Complete pg 93 - 94


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