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Lesson 8. Perimeter and Area. 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.

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lesson 8

Lesson 8

Perimeter and Area

perimeter
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.
example

C

D

A

B

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
circumference
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
example5
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.
example6
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

slide7
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 room’s floor.
  • Units used for area include square centimeters square miles, square yards, and acres.
area of a rectangle
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

example9

10

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:
altitudes of triangles

h

h

h

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:
area of a triangle
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).

example12

A

C

15

25

B

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

area of a parallelogram

h

b

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.
area of a trapezoid

b

h

B

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:
area of a circle
Area of a Circle
  • The area of a circle with radius r is found by multiplying pi by the radius squared.
  • The formula is
heron s formula
Heron’s Formula
  • Heron’s 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

adding areas
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.
subtracting areas
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.
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