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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 l.jpg

Lesson 8

Perimeter and Area


Perimeter l.jpg
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 l.jpg

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 l.jpg
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 l.jpg
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 l.jpg
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 room’s floor.

  • Units used for area include square centimeters square miles, square yards, and acres.


Area of a rectangle l.jpg
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 l.jpg

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 l.jpg

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 l.jpg
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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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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