11.3 Perimeters and Areas of Similar Polygons

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11.3 Perimeters and Areas of Similar Polygons

• Geometry
• Mrs. Spitz
• Spring 2006

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Objectives/Assignment

• Compare perimeters and areas of similar figures.
• Use perimeters and areas of similar figures to
  solve real-life problems.
• Assignment pp. 679-680 1-27 all

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Comparing Perimeter and Area

• For any polygon, the perimeter of the polygon is
  the sum of the lengths of its sides and the area
  of the polygon is the number of square units
  contained in its interior.

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Comparing Perimeter and Area

• In lesson 8.3, you learned that if two polygons
  are similar, then the ratio of their perimeters
  is the same as the ratio of the lengths of their
  corresponding sides. In activity 11.3 on pg.
  676, you may have discovered that the ratio of
  the areas of two similar polygons is NOT this
  same ratio.

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Thm 11.5 Areas of Similar Polygons

• If two polygons are similar with the lengths of
  corresponding sides in the ratio of ab, then the
  ratio of their areas is a2b2

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Thm 11.5 continued
kb
Side length of Quad I
a

ka
Side length of Quad II
b
I
II
Area of Quad I
a2

Area of Quad II
b2
Quad I Quad II
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Ex. 1 Finding Ratios of Similar Polygons

• Pentagons ABCDE and LMNPQ are similar.
• Find the ratio (red to blue) of the perimeters of
  the pentagons.
• Find the ratio (red to blue) of the areas of the
  pentagons

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10
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Ex. 1 Solution

• Find the ratio (red to blue) of the perimeters of
  the pentagons.
• The ratios of the lengths of corresponding sides
  in the pentagons is 510 or ½ or 12.
• The ratio is 12. So, the perimeter of pentagon
  ABCDE is half the perimeter of pentagon LMNPQ.

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10
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Ex. 1 Solution

• Find the ratio (red to blue) of the areas of the
  pentagons.
• Using Theorem 11.5, the ratio of the areas is 12
  22. Or, 14. So, the area of pentagon ABCDE is
  one fourth the area of pentagon LMNPQ.

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10
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Using perimeter and area in real life
Ex. 2 Using Areas of Similar Figures
Because the ratio of the lengths of the sides of
to rectangular pieces is 12, the ratio of the
areas of the pieces of paper is 12 22 or, 14
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Using perimeter and area in real life
Ex. 2 Using Areas of Similar Figures
Because the cost of the paper should be a
function of its area, the larger piece of paper
should cost about 4 times as much, or 1.68.
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Using perimeter and area in real life
Ex. 3 Finding Perimeters and Areas of Similar
Polygons

• Octagonal Floors. A trading pit at the Chicago
  Board of Trade is in the shape of a series of
  octagons. One octagon has a side length of about
  14.25 feet and an area of about 980.4 square
  feet. Find the area of a smaller octagon that
  has a perimeter of about 76 feet.

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Using perimeter and area in real life
Ex. 3 Solution

• All regular octagons are similar because all
  corresponding angles are congruent and
  corresponding side lengths are proportional.
• First Draw and label a sketch.

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Using perimeter and area in real life
Ex. 3 Solution

• FIND the ratio of the side lengths of the two
  octagons, which is the same as the ratio of their
  perimeters.

a
76
76
2
perimeter of ABCDEFGH

?


b
8(14.25)
114
3
perimeter of JKLMNPQR
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Using perimeter and area in real life
Ex. 3 Solution

• CALCULATE the area of the smaller octagon. Let A
  represent the area of the smaller octagon. The
  ratio of the areas of the smaller octagon to the
  larger is a2b2 2232, or 49.

Write the proportion.
?The area of the smaller octagon is about 435.7
square feet.
9A 980.4 4
Cross product property.
A 3921.6
Divide each side by 9.
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Use a calculator.
A ? 435.7
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Upcoming

• Quiz after 11.3. There are no other quizzes for
  this chapter.
• 11.4 Friday
• 11.5 Monday
• 11.6 Wednesday
• Test Friday, May 12