Section 10AFundamentals of Geometry

Pages 604-620

Perimeter and Area - Summary

10-A

Perimeter and AreaRectangles

10-A

Perimeter

= l+ w+ l+ w

= 2l + 2w

Area

= length × width

= l × w

Perimeter and AreaSquares

10-A

Perimeter

= l+l+l+l

= 4l

Area

= length × width

= l × l

= l2

Perimeter and AreaTriangles

10-A

Perimeter

= a + b + c

Area

= ½×b×h

Perimeter and AreaParallelograms

10-A

Perimeter

= l+ w+ l+ w

= 2l + 2w

Area

= length × height

= l×h

Perimeter and AreaCircles

10-A

Circumference(perimeter)

= 2πr

= πd

Area

= πr2

π ≈ 3.14159…

Practice with Area and Perimeter Formulas

Find the circumference/perimeter and area for each figure described:

43/617 A circle with diameter 16 centimeters

Circumference = πd = π×16 cm= 16π cm

Area = πr2 = π×(16/2 cm)2 = 64π cm2

10-A

16

Find the circumference/perimeter and area for each figure described:

51/617 A rectangular postage stamp with a length of 2.2 cm and a width of 2.0 cm

Perimeter = 2.2cm + 2.2cm + 2.0cm + 2.0cm = 8.4cm

Area = 2.2 cm × 2.0 cm = 4.4 cm2

10-A

Practice with Area and Perimeter Formulas

2

2.2

Find the circumference/perimeter and area for each figure described:

47/617 A square state park with sides of length 9 miles

Perimeter = 9 mi×4 = 36 miles

Area = (9 mi)2 = 81 miles2

10-A

Practice with Area and Perimeter Formulas

9

9

Find the circumference/perimeter and area for each figure described:

49/617 A parallelogram with sides of length 12 ft and 30 ft and a distance between the 30 ft sides of 6 ft.

Perimeter = 12ft +30 ft +12ft +30ft = 84ft

Area = 30ft × 6 ft = 180 ft2

10-A

Practice with Area and Perimeter Formulas

30

12 6

55/617 Find the perimeter and area of this triangle

Perimeter = 5+5+8 = 18 units

Area = ½ ×8×3 = 12 units2

10-A

Practice with Area and Perimeter Formulas

55

8

3

Applications of Area and Perimeter Formulas

57/617 A picture window has a length of 8 feet and a height of 6 feet, with a semicircular cap on each end (see Figure 10.20). How much metal trim is needed for the perimeter of the entire window, and how much glass is needed for the opening of the window?

59/618 Refer to Figure 10.14, showing the region to be covered with plywood under a set of stairs. Suppose that the stairs rise at a steeper angle and are 11 feet tall. What is the area of the region to be covered in that case?

61/618 A parking lot is bounded on four sides by streets, as shown in Figure 10.23. How much asphalt (in square yards) is needed to pave the parking lot?

Surface Area and Volume

10-A

89/619 Consider a softball with a radius of approximately 2 inches and a bowling ball with a radius of approximately 6 inches. Compute the surface area and volume for both balls.

10-A

Practice with Surface Area and Volume Formulas

Softball:Surface Area = 4xπx(2)2 = 16π square inchesVolume = (4/3)xπx(2)3 = (32/3) π cubic inches

Bowling ball:Surface Area = 4xπx(6)2 = 144π square inches

Volume = (4/3)xπx(6)3 = 288 π cubic inches

ex6/613 Which holds more soup – a can with a diameter of 3 inches and height of 4 in, or a can with a diameter of 4 in and a height of 3 inches?

10-A

Practice with Surface Area and Volume Formulas

Volume Can 1 = πr2h = π×(1.5 in)2×4 in = 9π in3

Volume Can 2 = πr2h = π×(2 in)2×3 in = 12π in3

Practice with Surface Area and Volume Formulas

69/618 The water reservoir for a city is shaped like a rectangular prism 250 meters long, 60 meters wide, and 12 meters deep. At the end of the day, the reservoir is 70% full. How much water must be added overnight to fill the reservoir?

Volume of reservoir = 250 x 60 x 12 = 180000 cubic meters

30% of volume of reservoir has evaporated.

.30 x 180000 = 54000 cubic meters have evaporated.

54000 cubic meters must be added overnight.

HomeworkPages 617-618#46,52,54,58,62,68,71

10-A

Section 10BProblem Solving

with Geometry

pages 621-637

For a right triangle with sides of length a, b, and c in which c is the longest side (or hypotenuse), the Pythagorean theorem states:

a2 + b2 = c2

a

b

c

Pythagorean Theorem

example If a right triangle has two sides of lengths 9 in and 12 in, what is the length of the hypotenuse?

(9 in)2+(12 in)2 = c2

81 in2+144 in2 = c2

225 in2= c2 9

12

c

2

225in c

15in = c

Pythagorean Theorem

example If a right triangle has a hypotenuse of length 10 cm and a short side of length 6 cm, how long is the other side?

6

10

b

b 64

b =8cm

(6)2 + b2 = (10)2

36 + b2 = 100b2 = (100-36) = 64

Pythagorean Theorem

ex5/626 Consider the map in Figure 10.30, showing several city streets in a rectangular grid. The individual city blocks are 1/8 of a mile in the east-west direction and 1/16 of a mile in the north-south direction. a) How far is the library from the subway along the path

shown?b) How far is the library from the subway “as the crow flies”

(along a straight diagonal path)?

library

subway

Pythagorean Theorem

ex6/626 Find the area, in acres, of the mountain lot shown below.

250 ft

1200 ft

ex9/629 You have 132 meters of fence that you plan to use to enclose a corral on a ranch. What shape should you choose if you want the corral to have the greatest possible area? What is the area of this optimized corral?

Optimization

87/634 Suppose you work for a company that manufactures cylindrical cans. Which will cost more to manufacture: a can with a radius of 4 inches and a height of 5 inches or a can with radius 5 inches and a height of 4 inches? Assume the cost of material for the tops and bottoms is $1.00 per square inch and the cost of material for the curved surface is $0.50 per square inch.

101/635 Telephone cable must be laid from a terminal box on the shore of a large lake to an island. The cable costs $500 per mile to lay underground and $1000 per mile to lay underwater. (See Figure 10.40/635) . As an engineer on the project, you decide to lay 3 miles of cable along the shore underground and then lay the remainder of the cable along a straight line underwater to the island. How much will this project cost? Your boss examines your proposal and asks whether laying 4 miles of cable underground before starting the underwater cable would be more economical. How much would your boss’s proposal cost? Will you still have a job?

Optimization

Two triangles are similar if they have the same shape (but not necessarily the same size), meaning that one is a scaled-up or scaled-down version of the other.

For two similar triangles:

• corresponding pairs of angles in each triangle are equal.Angle A = Angle A’, Angle B = Angle B’, Angle C = Angle C’

•the ratios of the side lengths in the two triangles are all equal

a’ b’

c’A’

B’

C’

a b

cA

B

C

a b c

a' b' c'

Similar Triangles

67/605 Complete the triangles shown below.

60

50 x y

40

10

Similar Triangles

Homework

Pages 633-635

#70, 88, 94, 96