Chapter 6: Perimeter, Area, and Volume

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Chapter 6 Perimeter, Area, and Volume

• Regular Math

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Section 6.1 Perimeter Area of Rectangles
Parallelograms

• Perimeter the distance around the OUTSIDE of a
  figure
• Area the number of square units INSIDE a figure

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Finding the Perimeter of Rectangles and
Parallelograms

• Find the perimeter of each figure.

• P S S S S
• P 26 20 26 20
• P 92 feet

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• P S S S S
• P 17.5x 11x 17.5x 11x
• P 57X units

• Find the perimeter of each figure.

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Using a Graph to Find Area

• Graph each figure with the given vertices. Then
  find the area of each figure.
• (-3, -1), (-3, 4), (1, 4), (1, -1)

• A bH
• b base H height
• A 4 X 5
• A 20 units squared

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Try this one on your own

• Graph each figure with the given vertices. Then
  find the area of the figure.
• (-4, 0), (2, 0), (4, 3), (-2, 3)

• A bH
• A 6 x 3
• A 18 units squared

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Finding Area and Perimeter of a Composite Figure

• Step One Fill in the missing sides.
• Step Two Solve for Perimeter
• Step Three Break the figure into rectangles.
• Step Four Solve for Area of each rectangle.
• Step Five Add the areas of each individual
  rectangles.

• Find the perimeter and area of the figure.

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Section 6.2 Perimeter and Area of Triangles and
Trapezoids

• Find the perimeter of each figure.

• P S S S
• P 22 22 27
• P 71 feet

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Try this one on your own

• Find the perimeter of each figure.
• P S S S
• P 2.5x 5y 2x 2x 4y
• P 6.5x 9y

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Find the area of triangles and trapezoids.

• Graph and find the area of each figure with the
  given vertices.
• (-1,-3), (0,2), (3,2), (3, -3)

• A ½ x h x (b1 b2)
• A ½ x 5 x (4 3)
• A ½ x 5 x (7)
• A 2.5 x 7
• A 17.5 units squared

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• A ½ x h x (B1 B2)
• A ½ x 3 x (3 5)
• A ½ x 3 x (8)
• A 1.5 x 8
• A 12 units squared

• Graph and find the area of each figure with the
  given vertices.
• (-3,-2), (-3,1), (0,1), (2, -2)

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Section 6.3 The Pythagorean Theorem
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Example 1 Finding the length of the hypotenuse.

• Find the length of the hypotenuse.

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• Graph the triangle with coordinates (6,1), (0,9),
  and (0,1).
• Find the length of the hypotenuse.

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Try this one on your own

• Find the length of the hypotenuse.
• C 6.40

• Graph the triangle with the following coordinates
  (1,-2), (1,7), and (13,-2).
• A 9
• B 12
• Find the length of the hypotenuse.
• C 15

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Example 2 Finding the length of a Leg in a Right
Triangle

• Solve for the unknown side in the right triangle.

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• Solve for the unknown side in the right triangle.
• b 24

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Example 3 Using the Pythagorean Theorem to Find
Area

• Use the Pythagorean Theorem to find the height of
  the triangle.
• Then, use the height to find the area of the
  triangle.

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Try this one on your own

• Use the Pythagorean Theorem to find the height of
  the triangle.
• h square root of 20 or 4.47
• Then, use the height to find the area of the
  triangle.
• A 17.89 units squared

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Section 6.4 Circles
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Finding the circumference of a Circle.

• Find the circumference of each circle, both in
  terms of pi and to the nearest tenth. Use 3.14
  for pi.
• Circle with radius 5 cm
• Circle with diameter 1.5 in

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Try these on your own

• Find the circumference of each circle, both in
  terms of pi and to the nearest tenth. Use 3.14
  for pi.

• Circle with radius 4 m
• C 8pi m or 25.1 m
• Circle with diameter 3.3 ft
• C 3.3pi or 10.4 ft

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Finding the Area of a Circle.

• Find the area of each circle, both in terms of pi
  and to the nearest tenth. Use 3.14 for pi.
• Circle with radius 5 cm
• Circle with diameter 1.5 in

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Try these on your own

• Find the area of each circle, both in terms of pi
  and to the nearest tenth. Use 3.14 for pi.

• Circle with radius 4 in
• A 16pi inches squared or 50.2 inches squared
• Circle with diameter 3.3 m
• A 2.7225pi meters squared or 8.5 meters squared

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Finding Area and Circumference on a Coordinate
Plane.

• Graph the circle with center (-1,1) that passes
  through (-1,3). Find the area and circumference,
  both in terms of pi and to the nearest tenth. Use
  3.14 for pi.
• Step One Graph Circle
• Step Two Find the radius
• Step Three Use the Area and Circumference Formula

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Try this one on your own

• Graph the circle with center (-2,1) that passes
  through (1,-1). Find the area and circumference,
  both in terms of pi and to the nearest tenth. Use
  3.14 for pi.
• A 9pi units squared and 28.3 units squared
• C 6pi units and 18.8 units

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• A bicycle odometer recorded 147 revolutions of a
  wheel with diameter 4/3 ft. How far did the
  bicycle travel? Use 22/7 for pi.
• The distance traveled is the circumference of the
  wheel times the number of revolutions.
• C pi(d) (22/7) (4/3) 88/21
• Circumference x Revolutions
• 88/21 x 147 616 feet

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Try this one on your own

• A Ferris wheel has a diameter of 56 feet and
  makes 15 revolutions per ride. How far would
  someone travel during a ride? Use 22/7 for pi.
• C 22/7(56) 176 feet
• Distance 176 (15) 2640 feet

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Section 6.5 Drawing Three Dimensional Figures

• Example 1 Drawing a Rectangular Box
• Use isometric dot paper to sketch a rectangular
  box that is 4 units long, 2 units wide, and 3
  units high.
• Step 1 Lightly draw the edges of the bottom
  face. It will look like a parallelogram.
• 2 units by 4 units
• Step 2 Lightly draw the vertical line segments
  from the vertices of the base.
• 3 units high
• Step 3 Lightly draw the top face by connecting
  the vertical lines to form a parallelogram.
• 2 units by 4 units
• Step 4 Darken the lines.
• Use solid lines for the edges that are visible
  and dashed lines for the edges that are hidden.

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Example 2 Sketching a One-Point Perspective
Drawing

• Step 1 Draw a rectangle.
• This will be the front face.
• Label the vertices A through D.
• Step 2 Mark a vanishing point V somewhere
  above your rectangle, and draw a dashed line from
  each vertex to V.
• Step 3 Choose a point G on line BV. Lightly
  draw a smaller rectangle that has G as one of its
  vertices.
• Step 4 Connect the vertices of the two
  rectangles along the dashed lines.
• Step 5 Darken the visible edges, and draw dashed
  segments for the hidden edges. Erase the
  vanishing point and all the lines connecting it
  to the vertices.

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Example 3 Sketching a Two-Point Perspective
Drawing

• Step 1 Draw a vertical segment and label it AD.
  Draw a horizontal line above segment AD. Label
  vanishing points V and W on the line. Draw dashed
  segments AV, AW, DV, and DW.
• Step 2 Label point C on segment DV and point E
  on segment DW. Draw vertical segments through C
  and E. Draw segment EV and CW.
• Step 3 Darken the visible edges. Erase horizon
  lines and dashed segments.

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Section 6.6 Volume of Prisms and Cylinders
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Example 1 Finding the Volume of Prisms and
Cylinders

• Find the volume of each figure to the nearest
  tenth.
• Step One Figure out what formula to use.
• Step Two Plug the numbers into the formula.
• Step Three Solve

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Try this one on your own

• Find the volume of each figure to the nearest
  tenth.

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Example 2 Exploring the Effects of Changing
Dimensions

• A juice can has a radius of 1.5 inches and a
  height of 5 inches. Explain whether doubling the
  height of the can would have the same effect on
  the volume as doubling the radius.

Original Double Radius Double Height

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Try this one on your own..

• A juice can has a radius of 2 inches and a height
  of 5 inches. Explain whether tripling the height
  would have the same effect on the volume as
  tripling the radius.

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Example 1 Finding the Volume of Prisms and
Cylinders

• Find the volume of each figure to the nearest
  tenth.
• A rectangular prism with base 1 meter by 3 meters
  and height of 6 meters

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Try these on your own

• Find the volume of each figure to the nearest
  tenth.
• A rectangular prism with base 2 cm by 5 cm and a
  height of 3cm

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Example 2 Exploring the Effects of Changing
Dimensions

• A juice box measures 3 inches by 2 inches by 4
  inches. Explain whether doubling the length,
  width, or height of the box would double the
  amount of juice the box holds.

Original Length Width Height

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Try this one on your own

• A juice box measures 3 inches by 2 inches by 4
  inches. Explain whether tripling the length,
  width, or height would triple the amount of juice
  the box holds.

Original Length Width Height

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Example 3 Construction Application

• Kansai International Airport is a man-made island
  that is a rectangular prism measuring 60 ft deep,
  4000 ft wide, and 2.5 miles long. What is the
  volume of rock, gravel, and concrete that was
  needed to build the island?

• Try this one on your own
• A section of an airport runway is a rectangular
  prism measuring 2 feet thick, 100 feet wide, and
  1.5 miles long. What is the volume of material
  that was needed to build the runway?

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Example 4 Finding the Volume of Composite Figures

• Find the volume of the milk carton.

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Try this one on your own

• Find the volume of the barn.

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Section 6.7 Volume of Pyramids and Cones
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Example 1 Finding the Volume of Pyramids and
Cones

• Find the volume of each figure.

• Try this one on your own
• Find the volume of each figure.

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Example 2 Exploring the Effects of Changing
Dimensions

• A cone has a radius 7 feet and height 14 feet.
  Explain whether tripling the height would have
  the same effect on the volume of the cone as
  tripling the radius.

Original Triple Height Triple Radius

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Try this one on your own

• A cone has a radius 3 feet and height 4 feet.
  Explain whether doubling the height would have
  the same effect on the volume as doubling the
  radius.

Original Double Height Double Radius

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Example 1 Finding the Volume of Pyramids and
Cones

• Find the volume of each figure.

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Try these on your own

• Find the volume of each figure.

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Example 3 Social Studies Application

• The Great Pyramid of Giza is a square pyramid.
  Its height is 481 feet, and its base has 756 feet
  sides. Find the volume of the pyramid.

• Try these on your own
• The pyramid of Kukulcan in Mexico is a square
  pyramid. Its height is 24 meters and its base has
  55 meter sides. Find the volume of the pyramid.

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Section 6.8 Surface Area of Prisms and Cylinders
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Example 1 Finding Surface Area

• Find the surface area of each figure.

• Try this one on your own

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• Try this one on your own
• Find the surface area of each figure.

• Find the surface area of each figure.

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Example 1 Finding Surface Area

• Finding the surface area of each figure.

• Try this one on your own
• Finding the surface area of each figure.

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Example 2 Exploring the Effects of Changing
Dimensions

• A cylinder has a diameter of 8 inches and a
  height of 3 inches. Explain whether doubling the
  height would have the same effect on the surface
  area as doubling the radius.

Original Double Height Double Radius

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Original Triple Radius Triple Height

• A cylinder has a diameter of 8 inches and a
  height of 3 inches. Explain whether tripling the
  height would have the same effect on the surface
  area as tripling the radius.

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Example 3 Art Application

• A web site advertises that it can turn your photo
  into an anamorphic image. To reflect the picture,
  you need to cover a cylinder that is 32mm in
  diameter and 100 mm tall with reflective
  material. How much reflective material do you
  need?

• Try this one on your own
• A cylindrical soup can has a radius of 7.6 cm and
  is 11.2 cm tall. What is the area of the label
  that covers the side of the can?

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Section 6.9 Surface Area of Pyramids and Cones
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Example 1 Finding Surface Area

• Find the surface area of each figure.

• Try this one on your own
• Find the surface area of each figure.

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• Try this one on your own
• Find the surface area of each figure.

• Find the surface area of each figure.

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Example 1 Finding Surface Area

• Try this one on your own
• Find the surface area of each figure.

• Find the surface area of each figure.

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Example 2 Exploring the Effects of Changing
Dimensions

• A cone has a diameter 8 in. and slant height 5
  in. Explain whether doubling the slant height
  would have the same effect on the surface area as
  doubling the radius.

Original Double Slant Height Double Radius

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Original Triple Radius Triple Slant Height

• A cone has diameter of 8 in. and slant height 3
  in. Explain whether tripling the slant height
  would have the same effect on the surface area as
  tripling the radius.

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Example 3 Life Science Application

• An ant lion pit is an inverted cone with the
  dimensions shown. What is the lateral surface
  area of the pit?

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• The upper portion of an hourglass is
  approximately an inverted cone with the given
  dimensions. What is the lateral surface area of
  the upper portion of the hourglass?

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Section 6.10 Spheres
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Example 1 Finding the Volume of a Sphere

• Find the volume of a sphere with a radius of 6
  ft, both in terms of pi and to the nearest tenth.

• Try this one on your own
• Find the volume of a sphere with radius 9 cm,
  both in terms of pi and to the nearest tenth.

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Example 2 Finding Surface Area of a Sphere

• Find the surface area, both in terms of pi and to
  the nearest tenth.

• Try this one on your own
• Find the surface area, both in terms of pi and to
  the nearest tenth.

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Example 3 Comparing Volumes and Surface Areas

• Compare the volume and surface area of a sphere
  with radius 21 cm with that of a rectangular
  prism measuring 28 x 33 x 42cm.

Sphere Volume Sphere Surface Area Prism Volume Sphere Surface Area

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Try this one on your own

• Compare volumes and surface areas of a sphere
  with radius 42 cm and a rectangular prism
  measuring 44 cm by 84 cm by 84 cm.

Sphere Volume Sphere Surface Area Prism Volume Prism Surface Area