Lesson 8

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# lesson 8 - PowerPoint PPT Presentation

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

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.
• Units for perimeter include inches, centimeters, miles, etc.

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
• 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
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.
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

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
• 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

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:

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
• 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).

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

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.

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
• 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 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