Perimeter and Area

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Section 10.2

• Perimeter and Area

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Exam III

• 8.3 - 3D shapes
• 9.1 - Congruence Transformations
• 9.2 - Symmetries and Tesselation
• 9.3 - Similarity (and Congruence)
• 10.2 - Perimeter (of anything) and Area (of
  triangles and quadrilaterals only)

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Perimeter and Area

• Perimeter is the distance along the border of a
  two dimensional shape
• Area of a shape is the measure of the
  (two-dimensional) space the shape takes up.

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Units

• The units for perimeter are units of distance
  (e.g. cm, m, ft, mi, yd, )
• The units for area are the square of units of
  distance (cm2, m2, ft2, mi2, yd2, ). Here m2 is
  read as square meters or meters squared.

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Perimeter of polygons

• Find the perimeter for the shape below

13in
6in
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Perimeter of polygons

• Find the perimeter for the shape below

7m
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Perimeter of polygons

• Estimate the perimeter of the shape below

4cm
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Draw this shape, and answer
How would you find the perimeter?
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What about this shape?
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Find the perimeter in terms of h and b
h
b
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Find the perimeter

• Assume the inner and outer rectangles are similar

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9
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Tools for measuring perimeter

• Ruler - increases numerical accuracy over
  eyeballing it.
• Straight edge - increases relative accuracy with
  respect to partial information.
• String - increases accuracy for the measurement
  of curves.

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Quadrilaterals

• The classification of quadrilaterals can be used
  to help us in making our task of measuring the
  perimeter easier.

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How much information do we need to find the
perimeter for
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Demystifying ?

• Before handing your students the formula for
  circumference, have them make connections first.

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Formulas can be dangerous

• Equating the task of finding the perimeter to
  plugging numbers into a formula is a common
  practice that completely obscures the purpose of
  a formula.
• Especially when that formula seems mysterious,
  like C 2?r to be explored later

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Shifting from perimeter to area

• Seconds, yards, degrees Celsius, quarts, and
  other standard units of measurement have everyday
  tools to measure them (stopwatch, yardstick,
  thermometer, measuring cup, etc.), but
  combination of units almost never have everyday
  tools to measure them (m2, foot-pounds, etc.)

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Shifting from perimeter to area (2)

• What are some (not necessarily mathematical)
  relationships between perimeter and area?

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Shifting from perimeter to area (2)

• What are some (not necessarily mathematical)
  relationships between perimeter and area?
• Formulas often use related numbers (radius,
  length, width, etc.)
• Both can be considered the defining
  characteristic of a shape. (e.g. space enclosed
  by a fence, sq. footage of a house, etc.)

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Measuring area

• When area is first introduced, why are the shapes
  almost always drawn on graph paper? Perimeter
  rarely is introduced by drawing shapes on graph
  paper, why area?

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Consider the shapes A and B below

• Jorge says A has a larger area
• Kate says B has a larger area
• Louis says their areas are equal
• Who is right?

B
A
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Geoboards and Area

• Assumption a 1 by 1 square on the geoboard has
  an area of 1 unit2

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Geoboard Find the area
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Geoboard and Area
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Geoboard and Area
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Geoboard and Area
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The process of measuring area

• How could you describe the process for
  determining the area of a triangle?

b
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Dissection

• How could you describe the process for
  determining the area of a triangle?

h
b
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Surround it

• How could you describe the process for
  determining the area of a triangle?

h
b
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The process of measuring area leads to the formula

• How could you describe the process for
  determining the area of a parallelogram?

h
b
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The process of measuring area leads to the formula

• How could you describe the process for
  determining the area of a parallelogram?

h
b
Dissection and rearranging.
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The process of measuring area leads to the formula

• How could you describe the process for
  determining the area of a parallelogram?

h
b
Dissection and rearranging.
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The process of measuring area leads to the formula

• How could you describe the process for
  determining the area of a trapezoid?

b2
h
b1
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The process of measuring area

• How could you describe the process for
  determining the area of a trapezoid?

b2
h
b1
Dissecting and using what we know of triangles
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The process of measuring area

• How could you describe the process for
  determining the area of a trapezoid?

b2
h
b1
Dissecting and using what we know of triangles
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Formulae

• Area for a rectangle is
• Area for a square is
• Area for a triangle is
• Area for a parallelogram is
• Area for a trapezoid is

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Area of a circle

• Area of a circle is A pr2
• Why?
• Draw a circle using a compass, and cut it out as
  carefully as possible.

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Area of an arc

• A 1/2 r2 ?
• Where ? is the radian measure of the angle at the
  center of the circle

• 1/2 p
• And we see
• The area is
• 1/4 of the
• Whole area.

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10.3

• Surface Area and Volume

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Surface Area

• Surface area of a 3 dimensional shape is
  analogous to the perimeter of a 2 dimensional
  shape
• It is the area of the boundary of the three
  dimensional object.

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Surface area of a polyhedron, cylinders, spheres,
and cones

• Now we can see a use for nets! (except for
  spheres)
• Surface area of a sphere of radius r is 4pr2
• It requires calculus to truly see this, but there
  are intuitive explanations (read 10.3 in the big
  book)

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Surface area of a cone is the trickiest of the
shapes we can examine through nets
4
6
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Surface area of a cone is the trickiest of the
shapes we can examine through nets
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6p
5
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Formula for a Cylinder
pr2
2pr
pr2
h
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Formula for a triangular prism
b
c
h
H
a
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