2-1) Find the area. (Composite Area Worksheet II # 1)

16 m4m

12 m

12 m

1-12) Hexagon ABCDEF is regular. Find the shaded area.

(Composite Area Worksheet I # 12)

4

2-4) Find the area. (Composite Area Worksheet II # 4)

10 2

2 2

4

4

2-8) Calculate the combined area in the right triangle and semi circle.

(Composite Area Worksheet II # 8)

15

6

1-2) Find the areas contained in the shapes. (Composite Area Worksheet I # 2)

4 cm

6 cm

6 cm

6 cm

8 cm

18 cm

2-6) Find the shaded area. The circles are both the same size and touch each other and the sides of the outer shape.

(Composite Area Worksheet II # 6)

12 ins

1-5) Find the areas contained in the shapes. (Composite Area Worksheet I # 5)

7 mm

6 mm

2 mm

12 mm

1-1) Find the areas contained in the shapes. (Composite Area Worksheet I # 1)

38 cm

20 cm

10 cm20 cm

2-2) Find the area. (Composite Area Worksheet II # 2)

15 in

5 in

5 in

10 in

5 in

2-14) Regular hexagon ABCDEF is inscribed in equilateral triangle XYZ. Find the shaded area.

(Composite Area Worksheet II # 14)

X F E Z

D

C

Y

B

A

6

6

6

2-3) Find the area. (Composite Area Worksheet II # 3)

6 in

8 in

9 in

5 in14 in

18 in

4 in

1-3) Find the area contained in the shapes. (Composite Area Worksheet I # 3)

20 ft

15 ft

20 ft

1-6) Find the areas contained in the shapes. (Composite Area Worksheet I # 6)

15 cm

12 cm

21 cm

27 cm

1-10) Find the shaded area. The circles are both the same size and touch each other and the sides of the outer shape.

(Composite Area Worksheet I # 10)

1-8) Find the areas contained in the shapes. (Composite Area Worksheet I # 8)

16 cm3 cm

3 cm

8 cm4 cm

12

20

5

2-11) Calculate the shaded area. (Composite Area Worksheet II # 11)

The main figure is a trapezoid. The main figure is a square. The unshaded part is a parallelogram and its base and height are half the lengths of the legs of the right triangle.

The main figure is a rectangle and the unshaded part is a rhombus.

The main figure is a parallelogram. The main figure is an isosceles triangle.

The main figure is a rectangle and the figure inside is totally symmetrical.

Find the area of this “strange” region.

9 3 20

The main figure is a square. The main figure is a rectangle and the unshaded part is a rhombus.

The unshaded part is a parallelogram and its base and height are half the lengths of the legs of the right triangle.

The top and bottom halves of the main figure are congruent and the white figures are parallelograms.

The main figure is a rectangle and its sides are bisected by the vertices of the unshaded figure.

The main figure is a trapezoid. Find the area of this “strange” region. The main figure is a rectangle.

17

8

2.5 2.5

9 10 2616

24

3

1 2 4

15

5

13

5

13 13

5 5

1313

10

7 4

3

12

10

3

10 2

14 4

10 6 11

2 2

3

7 14 50

6

5

33

5 5

5

12

16 12

11 12

20 8 9

12

16

13

18 16 5

8 10

4

13

14

20

15

15

6

6

5

Notes: Areas of Regular Polygons Name: ___________________________ Geometry 10-3 and 10-5 Date: ________________ Period: _____ In a regular polygon, a segment drawn from the center of the polygon perpendicular to a side of the polygon is called an apothem. In the figure at the right, CM is an apothem. A segment drawn from the center of the polygon to a vertex is called a radius of the polygon. In the figure at the right, CV is a radius. (It is also the radius of the circumscribed circle that encloses the regular pentagon.) C

V

M

The area of a regular polygon can be found by dividing the polygon into congruent isosceles triangles. For example, the pentagon can be divided into 5 triangles by drawing all five radii. If a regular polygon has n sides then it can be divided into n triangles.

Now find the area of one of the triangles (note: area of triangle = ½ bh). The height of the triangle will be the apothem. The base of the triangle is the length of one side, s, of the polygon. Therefore, area of the triangle = ½ sa. Since there are n triangles, multiply the area of the triangle by n to get the area of the polygon. Then, area of polygon = n [½ s·a] or ½ n·s·a

However, n·s is just the perimeter, P, of the polygon. Therefore, the area of a regular polygon with perimeter Pand apothem a is

A = ½ a P To use this formula, you need to know the length of a side and the apothem of the regular polygon. However, sometimes only the side length is given, but you can still find the apothem using these steps:

1. Draw a triangle with a vertex at the center 2. Draw the apothem (which splits the triangle into 2 right triangles) 3. Use trigonometry (or special right triangles):

θ = 360° ÷ n ÷ 2 opp = ½ side length adj = apothem (a) hyp = radius (most of the time you don’t know this or care about finding it)

4. Use the apothem and perimeter to find Area = ½ a P Page 547 Example 3: Find the area of a regular hexagon with 10-mm sides.

1. Draw a triangle with a vertex at the center 2. Draw the apothem 3. Use trig (or sp rt triangles):

θ = opp = adj = hyp =

4. Area = ½ a P =

Page 559 Example 1: Find the area of a regular pentagon with 8-cm sides.

1. Draw a triangle with a vertex at the center 2. Draw the apothem 3. Use trig (or sp rt triangles):

θ = opp = adj = hyp =

4. Area = ½ a P =

Extension: Area of Polygons Name: ___________________________ 10-3 Areas of Regular Polygons, and 10-5 Trigonometry and Area Date: ________________ Period: _____ Page 560 Ex 2: The Castel del Monte, built on a hill in southern Italy circa 1240, makes extraordinary use of regular octagons. One regular octagon, the inner courtyard, has radius 16 m. __________ Find the measure of the angle formed by the apothem and radius. θ = opp = adj = hyp =

__________ Find a, the apothem.

__________ Find x, which is half the length of a side.

__________ Find the perimeter of the courtyard.

__________ Find the area of the courtyard.

Page 561 Ex 3: The surveyed lengths of two adjacent sides of a triangular plot of land are 412 ft and 386 ft. The angle between the sides is 71°. __________ Find the height of the triangle. __________ Find the area of the plot.

Now, generalize this process to create a formula! Set up a trig ratio with ∠A. Solve for h. Plug it into the formula for area of a triangle.

Area of a Triangle Given SAS: (if the known angle is the included angle between the two known sides)

Area of ∆ABC = ( )Abc sin21

Page 561 Ex 3b: Two sides of a triangular building plot are 120 ft and 85 ft long. They include an angle of 85°. Find the area of the building plot to the nearest square foot. __________ Area of the plot

Equilateral triangles are sometimes easier to work with. Area of an equilateral triangle = 4

32s

Ex 5: Find the apothem, perimeter, and area of the equilateral triangle below. __________ Apothem __________ Perimeter __________ Area

Ex 6: Hexagons are made up of 6 congruent equilateral triangles. Find the apothem, perimeter, and area of a regular hexagon that has a side of 8 in. __________ Apothem __________ Perimeter __________ Area

16 a

x

386

71°

412

c

b A

B

C

a h

7

Areas of Irregular Polygons: Finding Area by Using Right Angles to Surround and to Dissect Figures Areas of polygons with coordinates for vertices can be found by surrounding the figure with a rectangle, by finding the area of the rectangle and the areas of the "extra" triangles, and by subtracting the areas of the triangles from the rectangle to find the area of the polygon. Ex 7: A house sits on a corner lot. Using the intersection of the streets as the origin, the vertices of the polygonal patio (in yards) are A(2,5), B(5,12), C(14,8), and D(9,1). The owner wants to determine the cost of adding a topcoat to the patio. The topcoat is to be two inches deep and costs $6.50 per cubic foot. How much will it cost for the topcoat to do this project?

12

10

8

6

4

2

B

C

A

5 10 15

D

Ex 8: Triangular parking lot ABC has vertices A(-7, 5), B(-2, -3), and C(5, 4). It is to be painted white to reflect the sun. If the paint is only available in gallon containers which cost $16.99 and cover 100 square feet, how much will this project cost? Will there be paint left over? If so, how much paint will be left over?

Areas of Composite Figures: To find the area of a composite figure, dissect it into the basic shapes, find the areas of the basic shapes, and add them together to find the area of the original shape. Ex 9: The floor of an auditorium is shaped like the drawing below. What is the minimum number of 8” by 8” tiles that are needed to tile the floor?

Ex 10: The top view of the floor of an office building is shown. It consists of part of a circle and two regular polygons. If the radius of the circle is 12 ft. and carpet costs $12 a square yard, $50 for removing the old carpet, and $75 for installing the new carpet, how much would it cost to recarpet the floor?

2

2

13'

20'

12'

Lab: Ratios of Perimeter vs. Area Name: ___________________________ 10-4 Perimeters and Areas of Similar Figures Date: ________________ Period: _____ You and a partner will explore the ratios of perimeter and area of similar figures made by dilating a rectangle*. On the graph, plot the vertices in the chart below and connect them. Find the rectangle’s perimeter and area. Then dilate the vertices by the scale factors listed in the chart below, and find the vertices, perimeters, and areas of the images. Finally, find the similarity ratio of the original to its image. Compare the perimeters and areas of the original and the image, and write the simplified ratios (from original to image) of their perimeters and areas. Vertices Perimeter Area Similarity

Ratio Ratio of

Perimeters Ratio of Areas

Original vertices of polygon (0, 0), (0, 2), (3, 2), (3, 0) 1 : 1 1 : 1 1 : 1

Dilate original by scale factor 2

1 : 2

Dilate original by scale factor 3

Dilate original by scale factor 4

* For a challenge, instead of a rectangle, dilate either a parallelogram with vertices (0, 0), (1, 2), (4, 2), (3, 0); a triangle with vertices (0, 0), (1, 2), (3, 0); or a kite with vertices (0, 0), (1, 1), (3, 0), (1, -1). You must know how to find lengths/distances on a coordinate plane, as well as how to simplify and add radicals. Note: If you are graphing the kite, change the x-axis so that it lies in the middle of the graph. How do the ratios of perimeters and the ratios of areas compare with the similarity ratios? Practice Exercises: 1. Two similar trapezoids have corresponding sides in the ratio 5 : 7.

__________ What is the ratio of their perimeters? __________ What is the ratio of their areas?

2. Two regular pentagons have side lengths of 4 cm and 10 cm. The area of the smaller pentagon is about 27.5 cm2. __________ What is the ratio of their sides? (Hint: remember to simplify!) __________ What is the ratio of their areas? __________ What is the area of the larger pentagon? (Hint: set up a proportion)

3. The areas of two similar triangles are 18 cm2 and 32 cm2. __________ What is the ratio of their areas? __________ What is the similarity ratio? (Hint: take the square roots) __________ What is the ratio of their perimeters?

Extension: What if only 1 dimension is dilated? For example, dilate the length only of a 2-by-3 rectangle by a scale factor of 2, so that the image is a 4-by-3 rectangle. What is the ratio of the perimeters? the areas? Does the same pattern from above apply?

Geometry Notes – Arc Length and Areas of Sectors and Segments of Circles A central angle is an angle whose vertex is the center of the circle. Arc measure = measure of its central angle.

Example 1: ∠BOC is a __________ __________.

m BC = m BD =

m ABC = is a _______________. ABC

m AB = m BAD =

AB is a __________ arc (less than _____). ABD is a __________ arc (greater than _____).

Arc length = 360m C where m is the measure of the central angle and C is the circumference.

Area of sector = 2

360m rπ where m is the measure of the central angle and r is the radius of the circle.

Example 2: a. Find the length of . b. Find the area of the shaded sector.ABC

Arc length = 120 (8)360

π Asector = 2120360

rπ

Arc length = 1 (8 )3

π Asector = 21 43π

Arc length = 83π units Asector =

163π units2

Example 3: Note: Sector of Circle – Triangle = Segment of Circle Given: P and m APB = 60˚ ∠

260 6360

π - 26 34

= 6 9 3π − units2

6

A

B

P

6

A

B

P

6

A

B

P- =

A

C

P

B

4

Given: P and m∠APC = 120˚

Challenge: Find the area of the shaded portion in each figure. The “dots” are centers of circles. 4. 5.

Arc Length, Sector Area, Segment Area Name_____________________________________ Geometry 10-6 and 10-7 Date_____________________Period____________ Find the shaded area. On problems 1-3, find the sector’s arc length also. Give answers in simplest exact form.

1. Asector = _____________ Arc length = __________

2. Asector = _____________ Arc length = __________

3. Asector = _____________ Arc length = __________

12

21

4. Asegment = _____________

5. Asegment = _____________

6. Asegment = _____________

120˚ 4

10. A track is formed around a football field by adding a semicircle to each end. How far will an athlete run if he makes one lap around the track (running on the inside of the lane)?

11. If the track lane is to be 4 feet wide, what will the area of the lane be? 14. The following four congruent are tangent and a square is created by joining the centers of the circles.

(The side of the square has measure 14.) Find the area of the shaded region. 15. Find the perimeter of the shaded region.

Answers: 10. (160 720) 1222.7ftπ + ≈ . 11. . 14. . 15. P 122880 656 4940.9ftπ+ ≈ 2A (196 49 )uπ= − 4 uπ= .

90˚

60˚

4

60˚

10 12

120˚

160 ft 360 ft

Goat on a Rope Name: ___________________________ Class Exercises Date: ________________ Period: _____ 1. If a goat is tied with a 10-foot rope to a stake in an open field, how much grazing area does he have?

2. The back of a 20 foot by 40 foot barn adjoins a 100 foot fence. If a goat is tied with a 16 foot rope to the fence post that joins the barn, how much grazing area in the barnyard does the goat have?

3. A dog is tied with a 6 foot rope to a corner of a building that is 15 feet by 15 feet. How much running area does the dog have?

4. A cow is tethered to a post alongside a barn 10 meters wide and 30 meters long. If the rope is 10 meters from a corner of the barn, and if the rope is 30 meters long, find the cow’s total grazing area to the nearest square meter. 5. Billy, a goat, has a rope of length 30 feet attached to his collar. If the other end of Billy's rope is attached to a "runner" that runs along the total perimeter (60 feet) of a barn in the shape of an equilateral triangle, and it can slide the entire length of the "runner", how much grazing room does Billy have? Neither Billy nor the rope can go in the barn.

Notes – 10.8 – Geometric Probability Length Probability Postulate: If a point on AB is chosen at random and C is between A and B, then the probability that the point is on AC is:

length AClength AB

Ex. What is the probability a point chosen at random on DI is also a part of: (a) EF (b) FI Ex. Elena’s bus runs every 25 minutes. If she arrives at her bus stop at a random time, what is the

probability that she will have to wait at least 10 minutes for the bus? (Hint: draw a timeline) Area Probability Postulate If a point in region A is chosen at random, then the probability that the point is in region B, which is in the interior of region A, is:

area of region Barea of region A

Ex. Joanna designed a new dart game. A dart in section A earns 10 points; a dart in section B earns 5 points; a dart in section C earns 2 points. Find the probability of earning each score. radius of circle A = 2 in. radius of circle B = 5 in. radius of circle C = 10 in. 11. The state of Connecticut is approximated by a rectangle 100 mi by 50 mi. Hartford is approximately

at the center of Connecticut. If a meteor hit the earth within 200 mi of Hartford, find the probability that the meteor landed in Connecticut.

D E F G H I

1 2 3 6 7 8 9 0 4 5

A B

C

Ex. Find the probability that a point chosen at random in this circle will be in the given section.

D

C B

A

E

50° 120°

40° 60°

(a) A (b) C (c) D

8 cm

Ex. Find the probability that a point chosen at random in each figure lies in the shaded region. Round your answer to the nearest hundredth. (a) (b) Regular hexagon (sides = 12)

inside a rectangle Ex. A dartboard is a square of radius 10 in. You

throw a dart and hit the target. Find the probability that the dart lies within 5 inches of the center of the square.

10. Roberto’s trolley runs every 45 minutes. If he arrives at the trolley stop at a random time, what is the probability that he will not have to wait more than 10 minutes?

12. A stop light at an intersection stays red for 60 seconds, changes to green for 45 seconds, and then turns yellow for 15 seconds. If Jamal arrives at the intersection at a random time, what is the probability that he will have to wait at a red light for more than 15 seconds?

Ex. Main Street intersects Martin Luther King Boulevard. The traffic lights on Main follow these cycles: green 20 s, yellow 5 s, red 50 s. As you travel along Main and approach the intersection, what is the probability that the first color you see is green?

Review of Exponents, Distributing, Factoring Name: ___________________________ (As it relates to area on the TAKS test) Date: ________________ Period: _____ Exponents: Multiplying, Dividing, and Raising to a Power * When in doubt, E X P A N D the exponent (x3 = x·x·x) and regroup or cancel pairs. 1. Which expression describes the area in a square

units of a rectangle that has a width of 4x3y2 and a length of 3x4y5?

2. Which expression represents the area of a triangle with a base of 2x2y4z units and a height of 5xy4z3 units?

3. The area of a rectangle is 30m11n5 square units. If the length of the rectangle is 6m4n2 units, how many units wide is the rectangle? (m ≠ 0 and n ≠ 0)

4. The area of a parallelogram is 40m9n7 square units. If the base of the parallelogram is 5m12n–3 units, how many units high is the parallelogram? (m ≠ 0 and n ≠ 0)

5. Write an expression which represents the area in square units of a square with a side length of

3 45x y ?

6. Write an expression which represents the volume in cubic units of a cube with a side length of 3 45x y . (Formula for volume of a cube is V = s3)

Write the rule for multiplying two exponents with the same base (for example, xa and xb). Write the rule for dividing two exponents with the same base (for example, xa and xb). Write the rule for raising an exponent to a power (for example, raising xa to the p-th power). Distributive Property and FOIL 7. What equation best represents the area, A, of the

rectangle below?

8. The area of the shaded portion of the rectangle shown below is 440 square feet. How can the area of the unshaded portion of the rectangle be expressed in terms of x in square feet?

9.

10. Tammy drew a floor plan for her kitchen, as shown below. Which expression represents the area of Tammy’s kitchen in square units?

11. Factor out the GCF (Greatest Common Factor) to write the expression below as the product of two factors: xx x 36484 23 −+

12. Factor out the GCF: xx x 364812 24 −+

Factoring to find the roots/solutions of equations like: 02 =++ cbxax* Find factors of a·c that add to b. 13. Factor x2 + 5x + 6.

14. Factor x2 + 6x – 27.

15. Factor x2 – 17x + 60.

16. Factor x2 – x – 42.

17. Factor 2x2 + x – 6.

18. Factor 4x2 – 19x + 12.

19. The area of a rectangle is given by the equation 185 , in which l is the rectangle’s length.

What is the length of the rectangle? 2 2 =− ll

20. The area of a rectangle is 814 , and the width is 4

3 2 ++ xx+x . What expression best descibes the

rectangle’s length?