Perimeter and Area

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Lesson 8

• Perimeter and Area

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Perimeter

• The perimeter of a closed figure is the distance
  around the outside of the figure.
• In the case of a polygon, the perimeter is found
  by adding the lengths of all of its sides. No
  special formulas are needed.
• Units for perimeter include inches, centimeters,
  miles, etc.

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Example

• In the figure, ABCD is a rectangle,
  and What is the perimeter of
  this rectangle?
• The opposite sides of a rectangle are congruent.
  So,
• So, the perimeter is

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Circumference

• The distance around the outside of a circle is
  traditionally called the circumference of the
  circle, not the perimeter.
• The circumference of a circle with diameter d is
  given by the formula
• Here, (pi) is a mathematical constant equal to
  approximately

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Example

• The radius of a circle is 5 inches.
• What is the circumference of this circle? Round
  to the nearest hundredth of an inch.
• First, note that
• So,
• Note scientific calculators have a
  button.

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Example

• In the figure, a rectangle is surmounted by a
  semicircle.
• Given the measurements as marked, find the
  perimeter of the figure.
• Note that the diameter of the circle is 10. So,
  the top and bottom sides of the rectangle are
  also 10.
• The left side of the rectangle is 15.
• The circumference of the semicircle is
• So, the perimeter of the figure is

5
10
15
15
10
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Area

• The area of a closed figure (like a circle or a
  polygon) measures the amount of space the
  figure takes up.
• For example, to find out how much carpet to order
  for a room, you would need to know the area of
  the rooms floor.
• Units used for area include square centimeters
  square miles, square yards, and acres.

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Area of a Rectangle

• The area of a rectangle is found by multiplying
  its base times its height (or length times
  width).
• If the base is b and the height is h as in the
  figure, then the area formula is
• Note that the product of two length units gives
  area units (like inches times inches equals
  square inches).

h
b
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Example

• The figure shown is a square whose diagonal
  measures 10.
• What is the area of the square?
• Using our knowledge of 45-45-90 triangles, note
  that each side of the square must be
• Now, since a square is a rectangle, we find its
  area by multiplying base times height

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Altitudes of Triangles

• The formula for the area of a triangle involves
  the length of an altitude of the triangle. So,
  first we discuss what an altitude is.
• An altitude is a line segment that runs from one
  vertex of the triangle to the opposite side or
  extension of the opposite side, and it is
  perpendicular to this opposite side (or
  extension).
• Some altitudes are drawn below and marked h

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Area of a Triangle

• The formula for the area of a triangle is
• where b is the length of the base (one of the
  sides of the triangle) and h is the height (the
  length of the altitude drawn to the base).

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Example

• The triangle in the figure is a right triangle
  with right angle at A, and sides as marked.
• Find the area of this triangle.
• We will take AB as the base. Then the height
  would be AC, which we can find with the
  Pythagorean Theorem
• So, the area is

20
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Area of a Parallelogram

• To find the area of a parallelogram multiply its
  base times its height.
• The base is any side of the parallelogram like
  the one marked b in the figure.
• The height is the length of an altitude drawn to
  the base like the one marked h in the figure.

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Area of a Trapezoid

• To find the area of a trapezoid multiply the
  height by the mean of the two bases.
• If the height is h and the bases are b and B as
  in the figure, then the area formula is

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Area of a Circle

• The area of a circle with radius r is found by
  multiplying pi by the radius squared.
• The formula is

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Herons Formula

• Herons Formula is used to find the area of a
  triangle when altitudes are unknown, but all
  three sides are known.
• If the lengths of the sides of the triangle are
  a, b, and c, then the area is given by the
  formula
• where s is the semiperimeter

b
a
c
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Adding Areas

• If you have to find the area of a complex shape,
  try dissecting the shape into non-overlapping
  simple shapes that you can find the area of.
  Then add the areas of the simple shapes.
• For example, note how the shape below is
  dissected into two rectangles.

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Subtracting Areas

• Sometimes the area of a complex figure,
  especially one with holes in it, can be found
  by subtracting the areas of simpler figures.
• For example, to find the shaded area below, we
  would subtract the area of the circle from the
  area of the rectangle.