Homework Section 11-1 - Saint Charles Preparatory School Links... · Chapter 11 Areas of Plane Figures ... sides of the polygon. Circumference of a Circle The perimeter of a circle

Chapter 11

Areas of Plane Figures

You MUST draw diagrams and show formulas for every applicable

homework problem!

Objectives

A. Use the terms defined in the

chapter correctly.

B. Properly use and interpret the

symbols for the terms and

concepts in this chapter.

C. Appropriately apply the postulates

and theorems in this chapter.

D. Correctly calculate the perimeters of various plane figures.

E. Correctly calculate the areas of various plane figures.

F. Correctly compare the areas of various plane figures.

G. Correctly calculate the areas of combined plane figures.

H. Correctly calculate linear and regional geometric probability.

Section 11-1

Areas of Rectangles

Homework Pages 426-427:

1-34

Excluding 18, and 30

Objectives

A. Understand the meaning of the

phrase ‘area of a polygon’.

B. Understand and apply the area of

a square postulate.

C. Understand and apply the area

congruence postulate.

D. Understand and apply the area

addition postulate.

E. Understand and apply the area of a rectangle theorem.

F. Apply the area theorems and postulates to real world problems.

G. Understand and apply the terms ‘altitude’, ‘base’, and ‘height’.

Reminder! Pattern Right Triangles

• Pattern right triangles can also be seen as RATIOS!

– The Pythagorean Triples (based on lengths of sides)

• 3x: 4x: 5x

• 5x: 12x: 13x

• 8x: 15x: 17x

• 7x: 24x: 25x

– The Special Right Triangles (based on angles)

• 45-45-90

– Based on sides 1x: 1x,

• 30-60-90

– Based on sides 1x: : 2x

x2

x3

What is LARGER?

• You have been cautioned not to refer to one figure being

LARGER than another based simply on:

– One side of a figure being longer than one side of a

second figure

– The measure of one angle of a figure being greater than

the measure of an angle of a second figure

– One figures ‘looking’ bigger than a second figure

• With the information in this chapter, you can finally say

which figure is LARGER!

• In geometry, the larger figure has the greater area.

• To be ‘technically correct’, we should refer to the ‘area of

a rectangular region’, not the ‘area of a rectangle’.

Base

Any side of a:

Triangle

Parallelogram

Rectangle

Rhombus

Square

or the parallel sides of a trapezoid.

The length of the base is indicated by a ‘b’.

Altitude any segment perpendicular to a line containing a base from a point on the opposite side of the figure.

Height length of an altitude, indicated by an ‘h’.

Key Measurement Terms

Perimeter The length of the ‘outside’ of a closed

geometric figure.

Perimeter of a Polygon The sum of the lengths of the

sides of the polygon.

Circumference of a Circle The perimeter of a circle

Key Measurement Terms

Terms in Action

Base

Base Base

Base

b = 6 Altitude

h = 4 Height

Base

Altitude

Height Altitude

Postulate 17

The area of a square is the square of the length of a side.

s

s

Area = s2

Postulate 18

If two figures are congruent, then they have the same area.

A

B C

D

E F

ABC DEF

Area of ABC = Area of DEF

So …

How would you correctly identify one figure being larger or

smaller than a second figure?

Postulate 19

The area of a region is the sum of the areas of its non-overlapping parts.

Area of the cross = red area + blue area + purple area

Theorem 11-1

The area of a rectangle equals the product of its base and its height.

b

h Area = (b)(h)

Sample Problems

x

y

If x = 4 and y = 6, then Area = ?

Using Theorem 11-1, the area is the product of the

base times height, 6 x 4 = 24 square units.

If the area = 70 square units and y = 4, then x = ?

Using Theorem 11-1, the area is the product of the

base times height, 70 square units = 4x. x = 70/4 = 35/2

If x = 2 and y = 7, then

perimeter = ?

Perimeter = 2 + 7 + 2 + 7 = 18

Sample Problems

1 3 5 7

b 12 cm 16 cm units (2x) units

h 5 cm units (x – 3) units

A 80 cm2

23

24

Questions 1 – 16 refer to rectangles. Complete the table where b is the

base of the rectangle, h is the height, A is the area, and p is the perimeter.

1. Area of a rectangle = ? Area of a rectangle = base x height A = 12cm x 5cm

60 cm2

3. Area of a rectangle = base x height 80cm2 = 16cm x h

5 cm

5. Area of a rectangle = base x height 3 2 4 2A units units

24 units2

7. Area of a rectangle = base x height A = (2x) units x (x – 3) units

(2x2 - 6x) units2

Sample Problems

9 11 13 15

b 9 cm 16 cm (a + 3) units x units

h 4 cm (a – 3) units

A (x2 - 3x) units2

p 42 cm

Questions 1 – 16 refer to rectangles. Complete the table where b is the

base of the rectangle, h is the height, A is the area, and p is the perimeter.

9. Area of a rectangle = base x height A = 9cm x 4cm

36 cm2

Perimeter of a rectangle = ? Perimeter of a rectangle = b + h + b + h = 2b + 2h

Perimeter of a rectangle = 2(9 cm) + 2(4 cm)

26 cm

11. Perimeter of a rectangle = 2b + 2h 42 cm = 2(16 cm) + 2(h cm)

42 cm = 32 cm + 2(h cm) 10 cm = 2(h cm)

5 cm

A = ?

80 cm2

13. A = (a + 3) units x (a – 3) units A = (a2 – 3a + 3a - 9) units2

(a2 - 9) units2

Perimeter of a rectangle = 2((a + 3) units) + 2((a – 3) units) p = (2a + 6 + 2a - 6) units

4a units

15. (x2 – 3x) units2 = (x) units x (h) units

(x – 3) units

p = 2(x units)+ 2((x – 3) units)

(4x – 6) units

Sample Problems

5

7

6

5

2

7

6

5

2

5

17.

7 x 5 = 35

6 x 5 = 30

5 x 5 = 25 5 x 2 = 10

6 x 5 = 30

By Postulate 19, Area = 35 + 30 + 25 + 10 + 30 = 130 sq units.

Consecutive sides of the figure are perpendicular. Find the area of each figure.

Sample Problems

10

26°

8

10

19.

21.


What type of triangle? Any patterns? ?: 8: 10 ?/2: 4: 5

6

Area = ? Area = 48 units2

What type of triangle?

Any patterns?

How would you find the height?

unitssin 26 units

10 units

height

units0.438371146 units

10 units

height

4.38371146 unitsheight

How would you find the base? units

cos 26 units10 units

base

units0.898794046 units

10 units

base

8.98794046 unitsbase area 8.98794046 units 4.38371146 units 2area 39.4 units

Sample Problems

8y

2x

3y

4x 3y

2x

3y

3y

23.


Sample Problems

25. Find the area of a square with diagonal of length d.

d

Definition of square:

• Parallelogram

• Congruent Sides

• 4 Right Angles

All squares are rhombi.

Diagonals of rhombi bisect angles.

45

45 90

Pattern Right Triangle 45-45-90 2,, xxx

unitssqddd

xxAread

xxd .2222

22

Sample Problems

27. A path 2 m wide surrounds a rectangular garden 20 m

long and 12 m wide. Find the area of the path.

29. A room 28 ft long and 20 ft wide has walls 8 ft high.

a. What is the total wall area?

b. How many gallon cans of paint should be bought to

paint the walls if 1 gal of paint covers 300 ft2?

31. A rectangle having an area of 392m2 is twice as long as it

is wide. Find its dimensions.

35. You have 40 m of fencing to enclose a pen for your dog.

If one side is x, what is the dimension, in terms of x, of the

other side? Express the area in terms of x. Find the area of

the pen if x = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. What are

the dimensions of the pen with the greatest area?

Section 11-2

Areas of Parallelograms, Triangles and

Rhombuses


1-36

Excluding 32, 34

Objectives

A. Understand and evaluate the area

of a parallelogram.

B. Understand and evaluate the area

of a triangle.

C. Understand and evaluate the area

of a rhombus by using the lengths

of the diagonals.

Theorem 11-2

The area of a parallelogram equals the product of a base and the

height to that base.

b

h Area = (b)(h)

Theorem 11-3

The area of a triangle equals half the product of a base and the height

to that base.

h

b

Area = (½)(b)(h)

Theorem 11-4

The area of a rhombus equals half the product of its diagonals.

Area = (½)(d1)(d2)

d1

d2

Sample Problems

12. Find the area of an equilateral triangle with

perimeter = 18.

a b

c

a = b = c

18 cba

onsubstitutiby 18 aaa

6183 aora3 3

6

60°

30° h

30-60-90 pattern right triangle

xxx 2,3,

Sample Problems

24. Find the area of an isosceles triangle with a 32 degree

vertex angle and a base of 8 cm. (find to the nearest 10th)

8

32

28.55

9519.1382

1

2

1

9519.132867.

4

4)2867(.

416tan

cm

hbArea

h

h

h

4

16

h

Sample Problems

27. If AM is a median of ABC, h = 5, and BC = 16 then

find the areas of ABC and AMB.

h

M B C

A

16

8 8

5

220)5)(8(2

1

2

1unitshbAMBArea

Sample Problems – Find the area of each figure.

1. A triangle with base 5.2 m and corresponding height 11.5 m. 1. What should you do first?

5.2 meters

11.5 meters

The area of a triangle equals half the product

of a base and the height to that base.

Supporting theorem?

Formula? Area = ½ (base x height)

Area = ½ (5.2 meters x 11.5 meters)

Area = 29.9 meters2

3. A parallelogram with base 3 2 and corrsponding height 2 2.3. What should you do first?

3 2 units

Supporting theorem?

The area of a parallelogram equals the product


Formula? Area = base x height

Area = 3 2 units 2 2 units


5. An equilateral triangle with sides 8 feet. 5. What should you do first?

8 ft. 8 ft.

8 ft.

The area of a triangle equals half the product


Supporting theorem?

Base = 8 feet. Height = ?

The height of a triangle is the length of its altitude.

The altitude of a triangle is the line segment

perpendicular to a line containing a base from a

point on the opposite side of the figure.

Isosceles Triangle Theorem – Corollary 3 The bisector of the vertex angle

of an isosceles triangle is perpendicular to the base at its midpoint.

4 ft.

Isosceles Triangle Theorem – Corollary 1 An equilateral triangle is also

equiangular.

60°

Pattern right triangle?

30 : 60 :90

x :x 3 : 2x

4 ft. across from 30°, 60° or 90° angle?

4 :x 3 : 2x

4 :4 3 :8

Height across from 30°, 60° or 90° angle?

Base = 8 ft. Height = 4 3 ft.

Area = ½ (base x height)

21 Area = 8 ft. 4 3 ft. =16 3 ft

2


7. 9. 15

5 13

5

6

8

5 10

10

7. What type of figure?

Supporting theorem?

The area of a parallelogram equals the

product of a base and the height to that base.

Any ideas?

5

6

8

5

10

10

Area = base x height

Area = 5 units x 8 units

Area = 40 units2

9. What type of figure?

So sorry, it is NOT a parallelogram!

Length?

12 Length?

9

Supporting theorem?

The area of a triangle equals half the

product of a base and the height to

that base.

Area = ½ (base x height) Area1 = ½ (5 units x 12 units) Area1 = 30 units2

Area2 = ½ (9 units x 12 units)

Area2 = 54 units2

Area = 30 units2 + 54 units2 = 84 units2


11. An isosceles right triangle with hypotenuse 8.

13. A parallelogram with a 45° angle and sides 6 and 10.

15. A 30°-60°-90° triangle with hypotenuse 10.

17. A rhombus with perimeter 68 and one diagonal 30.

Sample Problems

19. A square inscribed in a circle with radius r.

25. Find the areas of all three triangles.

2 8

Sample Problems

29a. Find the ratio of the areas of QRT and QTS.

29b. If the area of QRS is 240, find the length of the

altitude from S.

33. Find the area of an equilateral triangle with side 7.

35. The area of a rhombus is 100. Find the length of the two

diagonals if one is twice as long as the other.

h 24

12 18

Q

R T S

Section 11-3

Areas of Trapezoids


2-26 evens

Excluding 8, 14

Objectives

A. Correctly identify the bases and

height of a trapezoid.

B. Understand and apply the theorem

of the area of a trapezoid.

Theorem 11-5

The area of a trapezoid equals half the product of the height and the

sum of the bases.

h

b1

b2

Area = (½)(h)(b1 + b2)

Do you remember?

• What is the median of a trapezoid?

• What is the relationship between the lengths of the bases of a

trapezoid and its median?

m

212

1bbm

b1

b2

Median

Sample Problems

trapezoid 1.

b1 12

b2 8

h 7

A

m 217 12 8 70

2Area units

1 2

1

2Median b b

What should you do first?

12

7

8

Supporting theorem?

The area of a trapezoid

equals half the product of

the height and the sum of

the bases.

Supporting theorem for length of median?

Equation? 1 2

1

2Area h b b

The length of the median of a trapezoid is half the sum of its bases.

Equation? 1

12 8 102

Median units

Sample Problems

trapezoid 8.

b1

b2 3k

h 5k

A 45k2

m

1 15 unitsb k

1

15 3 9 units2

Median k k k


b1

5k

3k

45k2

Supporting theorem?

The area of a trapezoid

equals half the product of

the height and the sum of

the bases.

Equation? 1 2

1

2Area h b b

2 2

1

145 5 3

2k units k b k

2

190 (5 ) 3k k b k

118 3k b k

1 2

1

2Median b b

Supporting theorem for length of median?

The length of the median of a trapezoid

is half the sum of its bases.

Equation?

Sample Problems

17. An isosceles trapezoid with legs 10 and bases 10 and 22.

6 6

218 10 22 128

2Area units

By area addition: 21 16 8 10 8 6 8 128

2 2A units


10

22

10 10

Now what? Make familiar figures!

10

? ?

What type of triangles?

8

3x: 4x: 5x 6: 8 :10 for x = 2

8 Supporting theorem?

The area of a trapezoid equals half the

product of the height and the sum of

the bases.

Equation? 1 2

1

2Area h b b

Can you see another way of solving this problem?

Sample Problems

trapezoid 3. 5. 7.

b1 27 7

b2 9

h

A 90

m

6

13

3

14

5

31 9 2

36 2

Sample Problems

9. A trapezoid has area 54 and height 6. How long is its median?

11. 13.

15.

15

6

18

3

3

3

60° 60°

8

2

10

30°

Sample Problems

23. An isosceles trapezoid has bases 12 and 28. The area is

300. Find the height and the perimeter.

25. ABCDEF is a regular hexagon with side 12. Find the

areas of the three regions formed by diagonals AC and

AD.

27. A trapezoid with area 100 has bases 5 and 15. Find the

areas of the two triangles formed by extending the legs

until they intersect.

Section 11-4

Areas of Regular Polygons


1-20

Objectives

A. Correctly identify and apply the

terms ‘center of a regular

polygon’, ‘radius of a regular

polygon’, ‘central angle of a

regular polygon’, and ‘apothem of

a regular polygon’.

B. Understand and apply the theorem

regarding the area of a regular

polygon.

regular polygon: both equilateral and equiangular

Regular Reminder

90

9090

90

60 60

60

120 120

120

120120

120

center of a regular polygon: center of the circumscribed

circle.

radius of a regular polygon: distance from the center to a

vertex.

central angle of a regular polygon: the angle formed by

two radii drawn to consecutive vertices.

apothem of a regular polygon: distance from the center to

a side.

Parts of a Regular Polygon

Parts of a Regular Polygon

45

30

60

Center

Regular Polygon

Radius

Central Angle Apothem

9060

120

Theorem 11-6

The area of a regular polygon is equal to half the product of the

apothem and the perimeter.

Area = (½)(a)(p)

a

p = 4s

Area = (½)(a)(4s)

a

p = 6s

Area = (½)(a)(6s) a

p = 3s

Area = (½)(a)(3s)

Sample Problems – Complete the table for the regular polygon shown.

r a A

1. 28

45 45 90

: : 2x x x

1

2Area ap

2(8) 16

16 4 64

s

p


r 8 2

What is the measure of the central angle?

90°


Where is the apothem?

a

What type of triangle? Pattern?

Radius across from which angle?

8 2

x = ?

8 : 8 :

a = ?

8 units

45° 45°

Supporting theorem?

The area of a regular polygon is equal to half the

product of the apothem and the perimeter.

General equation for the area of regular polygon?

21(8)(64) 256

2Area units

Perimeter of the regular polygon?

Sample Problems

13. Find the area of an equilateral triangle with radius 34

2 3 unitsApothem

2(6) 12

3 3(12) 36

Side

Perimeter s u

21 1 2 3 36

36 32 2 1 1

Area ap units

What should you do first? Where is the radius?

What is the measure of the central angle?

4 3

4 3

4 3

120° 120°

120°

Where is the apothem?

60°


Pattern?

30 60 90

: 3 : 2x x x

Radius across from which angle?

4 3

x = ? 2 3x

2 3 : 6 :

Apothem = ?

1

2Area ap

Supporting theorem?

The area of a regular polygon is equal to half the

product of the apothem and the perimeter.

General equation for the area of regular polygon? Perimeter of the regular polygon?

Sample Problems

19. Find the apothem, perimeter and area of the inscribed polygon.

1

90

45a

ax

x

xxx

2

2

21

2,,

904545 22 2 units

2

4 4 2 units

s

p s

224

8

1

24

2

2

2

1

2

1unitsapA

Sample Problems

r a A

3. 49

r a p A

5. 6

7. 12

a r

a r

Sample Problems

r a p A

9. 4

11. 6 a r

15. A regular hexagon with perimeter 72.

21. Find the perimeter and the area of a regular dodecagon

inscribed in a circle with a radius of 1.

Section 11-5

Circumferences and Areas of Circles


2-30 evens

Excluding 4, 10, 14, 24, 28

Objectives

A. Understand and apply the term

‘circumference of a circle’.

B. Apply the equations for

circumference and area of a circle

to real world problems.

C. Understand the relationship

between the diameter of a circle

and its circumference.

D. Derive the approximate value for pi.

E. Identify multiple approximations for pi.

• circle:

– area: the limit of the sequence of numbers representing

the areas of the inscribed regular polygons as the

number of sides goes to infinity.

Area = r2

– circumference: the limit of the sequence of numbers

representing the perimeters of the inscribed regular

polygons as the number of sides goes to infinity.

Circumference = 2r or d

Area of a Circle

Area 1.3

Area = 2

Area 2.60

Area 2.83

Area r2 3.14

r = 1

Circumference of a Circle

Perimeter 5.20

Perimeter 5.66

Perimeter = 6

Perimeter 6.12

Circumference = 2 r

Circumference 6.28

r = 1

I’ll have a piece of

Key concepts concerning :

– represents the ratio of the circumference of a circle to

the diameter of that circle:

• = circumference / diameter (C = d)

– is a constant.

– is an irrational number.

– You can approximate as:

• 3.14

• 3.1416

• 3.14159

• 22/7

Sample Problems – Complete the table. Leave answers in terms of .

3.

r

C

A

2

5

2 22 5 2 22 5 22 5 110

2 1 units1 7 2 2 7 1 7 1 7

C r

2A r

2

2 222 5 (22)25 275 units

7 2 (7)4 14A r

Do you need an exact or estimated value?

Since the instructions state “Leave the answers in

terms of ”, you are finding an exact answer.

Equation for circumference of a circle in terms of

the radius?

5 2 5 2 52 5 units

2 1 1 2 2 1 1C

2C r

Equation for area of a circle in terms of the radius?

2

25 25 units

2 4A

What if you were asked to estimate the values? Which value of π would you use?

Since the radius is given as a fraction, it would be most logical to use π = 22/7 to

estimate the values.

Sample Problems

11. A basketball rim has a diameter 18 in. Find the

circumference and the area enclosed by the rim. Use π ≈ 3.14.

(3.14)(18) 56.52C d inches

2

2 218(3.14) 3.14 9 3.14 81 254.34

2A inches

Do you need an exact or estimated value?

Since the instructions state “Use ≈ 3.14”, you are finding an

estimated answer.

Equation for circumference of a circle in terms of the diameter?

C d

Equation for area of a circle in terms of the diameter? 2

2

dA

Sample Problems 13. If 6oz of dough are needed to make an 8 in pizza, how much

dough is needed to make a 16 in pizza of the same thickness?


8 inches 16 inches What is the area of each pizza?

2 2

2816 inches

2 2

dA

2 2

21664 inches

2 2

dA

Now what? Does the amount of dough needed grow proportionally?

2 2

6 ounces ounces

16 inches 64 inches

x

2

2

64 inches 6 ounces ounces

16 inchesx

2

2

64 inches 6 ounces ounces

16 inches 1x

ounces 4 6 ounces 24 ouncesx

Sample Problems

15. A school’s wrestling mat is a square with 40 ft sides. A circle 28 ft in

diameter is painted on the mat. No wrestling is allowed outside the circle.

Find the area of the mat not used during wrestling. Use 22/7.


SC

SC

40 feet

40

fee

t

28 feet

Exact or approximate answer?

Formula for area of entire wrestling mat? 2A s

2 240 feet 1600 feetA

Formula for area inside of the circle? 2

2

dA

2

2 222 28 2214 616 feet

7 2 7A

Formula for area outside of the circle?

outside square circleArea Area Area

2 2 2

outsideArea 1600 ft 616 ft 984 ft

Sample Problems

1. 5. 7.

r 7

C 20

A 25

9. Find the circumference and the area of a circle when

the diameter is (a) 42 (b) 14k. 722

Sample Problems

17. Which is a better buy, a 10 in pizza costing $5 or a 15 in

pizza costing $9?

21. The tires of a racing bike are approximately 70 cm in

diameter. (a) How far does a racer travel in 5 min if the

wheels are turning at a speed of 3 revolutions per second?

(b) How many revolutions does the wheel make in a 22

km race? Use 22/7.

Sample Problems

23. A target consists of four concentric circles with radii 1, 2,

3, and 4. (a) Find the area of the bull’s eye and of each

ring of the target. (b) Find the area of the nth ring if the

target contains n rings and a bull’s eye.

25. Find the area of each shaded region.

2 2

2

Sample Problems

4

4 27. Find the area of the blue region.

29. Draw a square and its inscribed and circumscribed circles.

Find the ratio of the areas of these two circles.

Section 11-6

Arc Lengths and Areas of Sectors


1-22

Excluding 20

Objectives

A. Identify the sector of a circle.

B. Calculate the sector area of a

circle based on arc length and

radius.

C. Calculate the radius of a circle

based on the arc length and sector

area.

D. Calculate the arc length of a circle based on the sector area and

radius.

E. Apply the sector area formulas to real world problems.

• sector of a circle: region of circle defined by two radii and the arc connecting the outer endpoints.

arc length equals the product of the circumference and the measurement of the arc divided by 360

sector area equals the product of the area and the measurement of the arc divided by 360

NOTE!

The MEASURE of the arc is the distance around the circle.

The measure of the arc equals the measure of the central angle and is expressed in degrees.

The LENGTH of the arc is the LINEAR distance of the arc.

The length is expressed in linear measures, such as feet or centimeters.

Arc Length & Sector Area

x3 6 0

If m = x , th e n th e le n g th o f = 2 rA B A B

2 x3 6 0

m = x , th e n th e a re a o f s e c to r A O B = rI f A B

r

x

A

B

If x = 90°, is it obvious that

you would walk ¼ of the

perimeter of the circle if you

started at point A and ended

at point B?

If x = 90°, is it obvious that

you would be taking ¼ of

the pizza if you sliced out

the red region?

Sample Problems

4.

m AOB

radius 3

240

2 3 4 units.360

Length AB

2 2240

Area of sector 3 6360

AO B units

240

Sector AOB is described by giving m AOB and the radius of circle O.

Make a sketch and find the length of and the area of sector AOB. AB


What is the equation/formula for the

length of an arc?

x3 6 0

If m = x , th e n th e le n g th o f = 2 rA B A B

240

3 O B

A

What is the equation/formula for the area of a sector?

2 x3 6 0

If m = x , th e n th e a re a o f s e c to r A O B = rA B

Does it make sense?

• Look at the previous problem. Does it make sense?

– If there are 360 degrees in a circle, then does it make

sense that the length of a 240 degree arc would be 2/3

of the circumference of the circle?

3

2

360

240

Circumference of a circle with radius 3

2 2(3) 6C r

240

2 2 3 4360 360

xLength AB r

2 4

3 6

Does it make sense?

• Look at the previous problem. Does it make sense?

– If there are 360 degrees in a circle, then does it make

sense that the sector area of a sector defined by a 240

degree arc would be 2/3 of the area of the circle?

3

2

360

240

22 2Area of circle with radius 3 3 9 unitsA r

22 2240

Area of sector 3 6360 360

xAO B r units

2

2

2 6

3 9

u

u

Sample Problems – find the area of the shaded region

15. 90°

8

8

2

2816

2circleA units

2

sec360

tor

xA r

214 4 8

2triangleA units

Find the area of the triangle.

4

4

216 4 8 12 8regionA u

Any ideas? Any familiar figures?

Equation/formula for area of a circle? 2

2circle

dA

Equation/formula for area of a sector?

2 2

sec

904 4

360torA units

Now what?

Equation/formula for area

of a triangle?

1

2triangleA bh

secregion circle tor triangleA A A A

Equation/formula for area the shaded region?

Sample Problems

11. The area of a sector AOB is 10 units2 and m AOB = 100°.

Find the radius of circle O.


O B

A 100°

10

2

sec360

tor

xA r

Equation/formula for area of a sector?

2 210010 units

360r

2 2510 units

18r

2 2510 units

18r

2 236 units r

6 unitsr

Sample Problems

1. 3. 5. 7. 9.

m AOB 30 120 180 40 108

radius 12 3 1.5

2

9 25

Sector AOB is described by giving m AOB and the radius of circle O.

Make a sketch and find the length of and the area of sector AOB. AB

Sample Problems

17.

60° 60°

120°

120°

2

13.

O

4

19. A rectangle with length 16 cm and width 12 cm is

inscribed in a circle. Find the area of the region inside the

circle but outside the rectangle.

23. A cow is tied by a 25 m rope to the corner of a barn. A

fence keeps the cow out of the garden. Find the grazing

area.

25. Two circles have radii 6 cm and their centers are 6 cm

apart. Find the area of the region common to both circles.

Sample Problems

Garden

Barn

15m

10m 10m

20m 25m

Section 11-7

Ratios of Areas


1-20

Objectives

A. Compare the areas of triangles

that share similar components.

B. Utilize the scale factor of similar

polygons to determine the ratios

of the perimeters of the similar

polygons.

C. Utilize the scale factor of similar polygons to determine the

ratios of the areas of the similar polygons.

D. Calculate perimeters and areas of similar polygons based on their

scale factor.

Do You Remember?

• What is a ‘scale factor’?

• The scale factor refers to similar polygons.

• For similar polygons, the scale factor is the ratio of the

lengths of two CORRESPONDING sides.

A

B C

D

E F

10 EF and 5, BC

DEF, ~ ABC If

2:1 10:5 factor scaleThen

Comparing the Areas of Triangles - Triangles With the Same Height

2

1

221

121

2

1

b

b

bh

bh

A

A

h

b2 b1

If two triangles have equal heights, then the ratio of

their areas equals the ratio of their bases.

Comparing the Areas of Triangles - Triangles With the Same Base

2

1

221

121

2

1

h

h

hb

hb

A

A

h1

h2

b

If two triangles have equal bases, then

the ratio of their areas equals the ratio

of their heights.

Comparing the Areas of Triangles - Similar Triangles

3b

3h

b

h

2

221

21

2

1

3

1

3

1

33

1

h3b3

hb

A

A

3

1 factor scale

If two triangles are similar, then the ratio of their areas is equal to

the square of their scale factor.

Theorem 11-7

If the scale factor of two similar figures is a : b, then

(1) the ratio of their perimeters is a : b

(2) the ratio of their areas is a2 : b2

ax

ax

bx

bx

b

a

bx

ax factor scale

b

a

bx4

ax4

bxbxbxbx

axaxaxax

blue perimeter

redperimeter

2

2

22

22

b

a

xb

xa

bxbx

axax

blue area

red area

Want to get these problems correct?

For any of these problems:

1. Make sure you are comparing similar figures.

2. Always setup the following table:

Scale Factor: a:b

Ratio of lengths

(radii, sides,

perimeters, etc):

a:b

Ratio of areas: a2:b2

Sample Problems

similar

figures

1.

scale

factor

1:4

ratio of

perimeters

ratio of

areas

Since the scale factor is 1:4,

then a = 1, b =4.

The ratio of the perimeters is

equivalent to the scale factor.

1:4

The ratio of the areas is equal to

the squares of the elements of the

scale factor.

1:16

a:b

a:b

a2:b2


a = ? b = ?

Are perimeters areas or lengths?

Ratio of the perimeters?

Ratio of the areas?

Sample Problems

similar

figures

8.

scale

factor

ratio of

perimeters

ratio of

areas

2:1

The ratio of the areas is 2:1.

2

2

then

2 or 2

1 or 1

a a

b b

1:2

1:2a:b

a:b

a2:b2


a = ? b = ?

Scale factor?

Ratio of perimeters?

Sample Problems

A B

C

D

E 8 6

5

13. Name two similar triangles and find the ratio of their

areas. Then find DE. What else do you know?

Do you know anything about the line segments

inside of the circle?

Theorem 9-11: When two chords intersect

inside a circle, the product of the segments of

one chord equals the product of the segments of

the other chord.

6 8 5DE 40 20

units6 3

DE Similar triangles?

ABE ~ DCE Why?

SAS Triangle Similarity – If 2 angles of 2 ’s are congruent, and the sides

including those angles are in proportion, then the ’s are similar.

Scale Factor: a:b

Ratio of lengths (radii,

sides, perimeters, etc): a:b


6:5

a = ? b = ?

a = 6 b = 5

6:5

36:25 Scale factor?

Ratio of areas?

Sample Problems

15. A quadrilateral with sides 8 cm, 9 cm, 6 cm, and 5 cm has an area of 45 cm2.

Find the area of a similar quadrilateral whose longest side is 15 cm.

LOGIC Since the longest side of the first quadrilateral is 9 cm, and

the quadrilaterals are similar, then the longest side of the second

quadrilateral must CORRESPOND to the side of length 9 cm.

Scale factor 9:15 or 3:5

a = 3, b = 5

2 2 2 2Ratio of areas : 3 :5a b

9 45

25 x


8

5

6

9

15

Scale Factor: a:b

Ratio of lengths (radii,

sides, perimeters, etc):

a:b


Scale factor?

3:5

a = ? b = ?

Ratio of areas?

9:25

2125x cm

Sample Problems

similar

figures

3. 5. 7.

scale

factor

r:2s

ratio of

perimeters

3:13

ratio of

areas

9:64

Sample Problems

9. On a map of California, 1 cm corresponds to 50 km. Find

the ratio of the map’s area to the actual area of California.

11. L, M and N are the midpoints of the sides of ABC. Find

the ratio of the perimeters and the ratio of the areas of LMN

& ABC.

Sample Problems

Find the ratios of the areas of (a) I & II, (b) I & III

I II

III

10

8

9 7

17. 19.

I

II

III

8

6

9 9

Sample Problems

A B

C D E

F G

4 6

2

21. In the diagram, ABCD is

a parallelogram. Name four

pairs of similar triangles and

give the ratio of the areas for

each pair.

A B

C D

F

E

4 8

9

12

23. ABCD is a parallelogram.

ABF of area

DEF of area

CEB of area

DEF of area

DEBA trapof area

DEF of area

Sample Problems

I II

III

IV

8

18

25. Find the ratio of the areas

(a) I & III (b) I & II

(c) I & IV (d) II & IV

(e) I & the trapezoid

27. Find the ratio of the areas

of region I and II.

I

II

3 5

2

Section 11-8

Geometric Probability


1-16

Excluding 10, 12

Objectives

A. Understand the classical meaning

of probability.

B. Understand and apply the concept

of linear probability.

C. Understand and apply the concept

of regional (area) probability.

“Classical” Probability

• Classical probability depends on:

– Knowing all possible outcomes to an experiment.

– Each possible outcome having the same chance of

occurring.

– Knowing the number of outcomes that describe the

‘success’ of an event.

• If I flip a coin:

– The possible outcomes are ‘heads’ or ‘tails’.

– If the coin in ‘fair’, then it is equally likely that a ‘head’

will appear as a ‘tail’.

– A successful event might be flipping the coin and

having it come up tails.

“Classical” Probability

The probability of an event occurring is:

outcomes possible'' ofNumber

outcomes 'successful' ofNumber

In the coin experiment, a successful outcome would be the

appearance of a tail when the coin is flipped.

The possible outcomes would be a ‘head’ or a ‘tail’.

2

1

outcomes # Total

successes of # (Tails)y Probabilit

The probability of drawing a heart from a standard deck of cards:

4

1

52

13

outcomes # Total

successes of # (Heart)y Probabilit

geometric probability:

– linear probability: the chances of a randomly chosen

point being on a given segment. Found by dividing the

length of the target segment by the length of the total

segment.

– regional probability: the chances of a randomly chosen

point being within the boundary of a given region.

Found by dividing the area of the target region by the

total area of the object.

Linear Geometric Probability

A C B

ABAC=ACon is Ppoint y that Probabilit

Regional Geometric Probability

S of areaR of area = Rregion in is Ppoint a that obabilityPr

R S

Sample Problems

3. A friend promises to call you at home sometime between 3

PM and 4 PM. At 2:45 PM you must leave your house

unexpectedly for half an hour. What is the probability that

you miss the first call?

2:45 3:00 3:15 4:00

Gone Call

Miss

15 minutes

60 minutes

4

1

minutes 60

minutes 15 (Miss)y Probabilit

Sample Problems

7. A dart is thrown at a board 12 m long and 5 m wide. Attached to the board are 30 balloons, each with radius 10 cm. Assuming each balloon lies entirely on the board, find the probability that a dart that hits the board will also hit a balloon.

1200 cm

500 cm

222 10010 cmrAballoon

22

30 300010030 cmunitsA balloons

21200 500 600,000boardA bh cm

2

2

3000PR (balloon hit)

600,000 200

cm

cm

Sample Problems

1. If M is the midpoint of AB and Q is the midpoint of MB.

If a point on AB is picked at random, what is the

probability that the point is on MQ?

5. A circular dartboard has a 40 cm diameter. Its bull’s eye

has a diameter 8 cm. (a) If an amateur throws a dart and

hits the board, what is the probability that the dart hits the

bull’s eye? (b) After many throws, 75 darts have hit the

target. Estimate the number hitting the bull’s eye.

9. At a carnival game, you can toss a quarter on a large table

that has been divided into squares 30 mm on a side. If the

coin comes to rest without touching any line, you win.

Otherwise you lose your coin. What are your chances of

winning on one toss if a quarter has a radius of 12 mm.

Sample Problems

11. A piece of wire 6 in long is cut into two pieces at a

random point. What is the probability that both pieces of

wire will be at least 1 in long?

13. Darts are thrown at a 1 m square which contains an

irregular red region. Of 100 darts thrown, 80 hit the

square. Of these, 10 hit the red region. Estimate the area

of the region.

15. Find the radius (r) of a coin used in the carnival game

described in problem #9 if the the probability of winning

is 0.25 and the formula for calculating this probability is

2

30

r230 winningof yprobabilit

Chapter 11

Areas of Plane Figures

Review

Homework Page 471: 2-18 evens

Documents

Homework Section 11-1 - Saint Charles Preparatory School Links... · Chapter 11 Areas of Plane Figures ... sides of the polygon. Circumference of a Circle The perimeter of a circle