Section 10AFundamentals of Geometry

Pages 578-588

Perimeter and Area - Summary

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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

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Perimeter

= l+ w+ l+ w

= 2l + 2w

Area

= length × height

= l×h

Perimeter and AreaCircles

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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:

33/589 A circle with diameter 25 centimeters

Circumference = πd = π×25 cm= 25π cm

Area = πr2 = π×(25/2 cm)2 = 156.25π cm2

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25

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

41/589 A rectangle with a length of 2 meters and a width of 8 meters

Perimeter = 2m + 2m + 8m + 8m = 20 meters

Area = 2 meters × 8 meters = 16 meters2

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Practice with Area and Perimeter Formulas

2

8

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

37/589 A square with sides of length 12 miles

Perimeter = 12 km×4 = 48 miles

Area = (12 meters)2 = 144 miles2

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Practice with Area and Perimeter Formulas

12

12

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

39/589 A parallelogram with sides of length 10 ft and 20 ft and a distance between the 20 ft sides of 5 ft.

Perimeter = 10ft +20 ft +10ft +20ft = 60ft

Area = 20ft × 5 ft = 100 ft2

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Practice with Area and Perimeter Formulas

20

10 5

45/589 Find the perimeter and area of this triangle

Perimeter = 9+9+15 = 33 units

Area = ½ ×15×4 = 30 units2

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Practice with Area and Perimeter Formulas

99

15

4

Applications of Area and Perimeter Formulas

47/589 A picture window has a length of 4 feet and a height of 3 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?

49/589 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 14 feet tall. What is the area of the region to be covered in that case?

51/589 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

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79/591 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.

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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

ex6/585 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

59/585 The water reservoir for a city is shaped like a rectangular prism 300 meters long, 100 meters wide, and 15 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 = 300 x 100 x 15 = 450000 cubic meters

30% of volume of reservoir has evaporated.

.30 x 450000 = 135000 cubic meters have evaporated.

135000 cubic meters must be added overnight.

HomeworkPages 589-590# 34, 52, 58, 61, 84

10-A

Section 10BProblem Solving

with Geometry

pages 597-608

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

ex5/597 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

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 603-605

#52,66,84,86,88