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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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Presentation Transcript

Lesson 8

Perimeter and Area

• 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.

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

• 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

• 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.

• 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

• 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.

• 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

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

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:

• 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).

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

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.

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:

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

• The formula is

• 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

• 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.

• 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.