Section 9'3: Perimeter and Area

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Section 9.3 Perimeter and Area

• Dr. Fred Butler
• Math 121 Fall 2004

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Perimeter

• The perimeter P of a two-dimensional figure is
  the sum of the lengths of the sides of the
  figure, measured in the same units as the lengths
  of the sides (for example feet, or inches).
• If the figure to the right has the side lengths
  as labeled, the perimeter P 3434519.

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Area

• The area A of a figure is the region within the
  boundaries of the figure, measured in square
  units (for example if the sides of the figure are
  measured in inches, the area will be measured in
  inches2).
• The rectangle pictured to the right has area 40
  square units.

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Perimeter and Area of a Rectangle

• The perimeter of a rectangle pictured below with
  side lengths l and w is
• P l w l w 2l 2w.
• The area of a rectangle as below is
• A l x w.

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PRS Question 4.12

• What is the perimeter and area, respectively, of
  the rectangle below?
• 1. P60, A34 2. P34, A60
• 3. P34, A120

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Area of a Square

• A square is a rectangle with four equal sides, so
  the length equals the width.
• If we call both the length and the width s, then
  using the formula
• A l x w
• we get that the area of a square is given by
• A s x s s2.

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Area of a Parallelogram

• If we were to cut off the red part of the
  parallelogram and attach it to the right side of
  the figure, the result would be a rectangle with
  sides b and h.
• So the area of a parallelogram with base b and
  height h is
• A b x h.

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Area of a Triangle

• Given a triangle with base b and height h, we put
  a second identical triangle on top to make a
  parallelogram with base b and height h.
• Since this parallelogram has area bxh, we see the
  area of the original triangle is given by
• A ½ bh.

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Area of a Trapezoid

• If we draw the diagonal as shown, we see that a
  trapezoid is made up of two triangles, one with
  base b1 and height h (the green one), the other
  with base b2 and height h (the orange one).

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Area of a Trapezoid contd.

• The area of the green triangle is 1/2b1h, and the
  area of the orange triangle is 1/2b2h.
• Thus the area of a trapezoid is the sum of the
  areas of the two triangles 1/2b1h 1/2b2h, which
  is
• A 1/2h(b1b2).

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Area and Perimeter (Draw Shapes on Board)
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Pythagorean Theorem

• The Pythagorean Theorem in words says The sum of
  the squares of the lengths of the legs of a right
  triangle equals the square of the length of the
  hypotenuse.
• Symbolically, if a, b, and c represent the length
  labeled on the triangle pictured, then
• a2 b2 c2.

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Circles

• A circle is a set of points that are all the same
  distance from a fixed point called the center.
• A radius of a circle, denoted r, is a line
  segment from the center to any point on the
  circle.
• A diameter of a circle, denoted d, is a line
  segment through the center or the circle with
  both endpoints on the circle.
• Note that d 2xr.

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PRS Question 4.13

• If a and b are natural numbers, then the length
  of the hypotenuse is always an irrational number.
• 1. True 2. False

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Circumference and Area

• The circumference of a circle is the length
  around the outside of the circle.
• Circumference is given by the formula
• C 2pr.
• The area of a circle with radius r is given by
• A pr2.

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Comparing Pizzas

• Suppose the large pizza at Pizza Place A has a 16
  in. diameter and sells for 15, and the large
  pizza at Pizza Place B has a 12 in. diameter, and
  sells for 10.
• Which pizza is the better deal?

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Comparing Pizzas contd.

• If we can find the area of each pizza, then we
  divide the total cost for the pizza by the area
  of the pizza to get the cost per square inch.
• We need to use the formula Apr2.
• Note that r8 in the first pizza, are r6 in the
  second.

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Find Pizza Areas

• So the area of the first pizza with r8 is
• A p(8)2 (3.14)(64) 200.96 in2,
• and the are of the second pizza with r6 is
• A p(6)2 (3.14)(36) 113.04 in2.
• So the cost per square inch for the first pizza
  is 15/(200.96 in2) and the cost per square inch
  for the second pizza is 10/(113.04 in2).

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PRS Question 4.14

• Which pizza is the better deal?
• 1. Pizza A 2. Pizza B

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Lecture Summary

• The perimeter of a two-dimensional figure is the
  sum of the lengths of the sides of the figure.
• The area of a figure is the region within the
  boundaries of the figure, measured in square
  units.
• We discussed formulas for the perimeter and area
  of a rectangle, square, parallelogram, triangle,
  and trapezoid.

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Lecture Summary contd.

• The Pythagorean Theorem says that for the right
  triangle pictured,
• a2 b2 c2.
• The circumference of a circle with radius r is
  2pr, and the area is pr2.

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Homework

• Do problems from Section 9.3 of textbook.
• Homework quiz 4 is due Wednesday, November 10 by
  11 PM Web CT time.
• No class Thursday!
• Study for Exam 4!