2-1) Find the area.
(Composite Area Worksheet II # 1)
16 m
(Composite Area Worksheet II # 4)
2
(Composite Area Worksheet I # 12)
2
2
12 m
12 m
2-6) Find the shaded area. The circles
are both the same size and touch each
other and the sides of the outer shape.
6
10
4
4
4 cm
(Composite Area Worksheet II # 8)
4
4m
1-2) Find the areas contained in the
shapes. (Composite Area Worksheet I # 2)
2-8) Calculate the combined area
in the right triangle and semi circle.
2-4) Find the area.
1-12) Hexagon ABCDEF is
regular. Find the shaded area.
15
1-5) Find the areas contained in the
shapes. (Composite Area Worksheet I # 5)
1-1) Find the areas contained in the
shapes. (Composite Area Worksheet I # 1)
7 mm
(Composite Area Worksheet II # 6)
2 mm
6 cm
18 cm
6 cm
12 ins
8 cm
20 cm
20 cm
6 mm
6 cm
38 cm
12 mm
2-2) Find the area.
(Composite Area Worksheet II # 2)
15 in
2-14) Regular hexagon ABCDEF is
inscribed in equilateral triangle
XYZ. Find the shaded area.
5 in
4 in
8 in
5 in
6
6
10 in
6
X
1-6) Find the areas contained in the
shapes. (Composite Area Worksheet I # 6)
15 cm
(Composite Area Worksheet II # 3)
Y
5 in
1-3) Find the area contained in the
shapes. (Composite Area Worksheet I # 3)
2-3) Find the area.
(Composite Area Worksheet II # 14)
5 in
B
14 in
C
A
6 in
D
15 ft
F
E
Z
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)
18 in
1-8) Find the areas contained in the
shapes. (Composite Area Worksheet I # 8)
16 cm
2-11) Calculate the shaded area.
(Composite Area Worksheet II # 11)
20
3 cm
3 cm
12
8 cm
27 cm
20 ft
20 ft
9 in
12 cm
21 cm
10 cm
4 cm
5
The main figure is a trapezoid.
The main figure is a square.
9
3
8
9
The unshaded part is a parallelogram and
its base and height are half the lengths
of the legs of the right triangle.
16
3
26
10
The main figure is a rectangle and the
unshaded
part is a
20
rhombus.
24
17
1
2
The main figure is a parallelogram.
2.5
4
2.5
The main figure is an isosceles
triangle.
15
13
13
7
13
The main figure is a rectangle and
the figure
inside is
totally
symmetrical.
12
4
13
5
10
5
3
10
5
2
8
10
14
4
3
6
11
12
2
3
2
The main figure is a rectangle and the
unshaded
part is a
16
rhombus.
7
9
10
13
5
The main figure is a square.
Find the area of this “strange” region.
The unshaded part is a parallelogram and
its base and height are half the lengths
of the legs of the right triangle.
14
11
50
20
The top and bottom halves of the main
figure are congruent and the white
figures are parallelograms.
3
3
12
5
5
5
5
6
12
The main figure is a
rectangle and its sides
are bisected
16
by the
vertices
of the
unshaded
figure.
12
The main figure is a trapezoid.
6
Find the area of this “strange” region.
16
18
15
5
The main figure is a rectangle.
5
10
8
14
6
13
4
15
13
20
Notes: Areas of Regular Polygons
Name: ___________________________
Date: ________________ Period: _____
Geometry 10-3 and 10-5
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.)
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.
V
C
M
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 P
and apothem a is
A=½aP
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
10-3 Areas of Regular Polygons, and 10-5 Trigonometry and Area
Name: ___________________________
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 =
16
a
x
__________ 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°.
386
__________ Find the height of the triangle.
__________ Find the area of the plot.
71°
412
Now, generalize this process to create a formula!
B
Set up a trig ratio with ∠A.
c
h
a
Solve for h.
C
A
b
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)
1
Area of ∆ABC = bc(sin A)
2
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.
Ex 5: Find the apothem, perimeter, and area of
the equilateral triangle below.
__________ Apothem
Area of an equilateral triangle =
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
__________ Perimeter
__________ Area
__________ Area
7
s2 3
4
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?
B
12
10
C
8
6
A
4
2
2
D
2
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?
5
10
15
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?
20'
13'
12'
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?
Lab: Ratios of Perimeter vs. Area
Name: ___________________________
Date: ________________ Period: _____
10-4 Perimeters and Areas of Similar Figures
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
Original vertices
of polygon
Dilate original by
scale factor 2
Dilate original by
scale factor 3
Dilate original by
scale factor 4
Perimeter
Area
(0, 0), (0, 2), (3, 2), (3, 0)
Similarity
Ratio
Ratio of
Perimeters
Ratio of
Areas
1:1
1:1
1:1
1:2
* 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 __________ __________.
p =
m BC
p =
m BD
mq
ABC =
q
ABC is a _______________.
mp
AB =
q =
m BAD
p
AB is a __________ arc (less than _____). q
ABD is a __________ arc (greater than _____).
m
Arc length =
360
Area of sector =
C
where m is the measure of the central angle and C is the circumference.
m
360
πr2
where m is the measure of the central angle and r is the radius of the circle.
q.
a. Find the length of ABC
Example 2:
Given: : P and m ∠ APC = 120˚
P
A
Arc length =
120
π (8)
360
Arc length =
Arc length =
4
C
b. Find the area of the shaded sector.
Asector =
120
πr 2
360
1
(8π )
3
Asector =
1
π 42
3
8π
units
3
Asector =
16π
units2
3
B
Example 3:
Given: : P and m ∠ APB = 60˚
P
Note: Sector of Circle – Triangle = Segment of Circle
P
-
6
6
6
B
B
B
A
A
A
60
π 62
360
P
=
-
62 3
4
=
6π − 9 3 units2
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
Geometry 10-6 and 10-7
Name_____________________________________
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 = _____________
2. Asector = _____________
3. Asector = _____________
Arc length = __________
Arc length = __________
Arc length = __________
21
12
120˚
4
60˚
90˚
4. Asegment = _____________
5. Asegment = _____________
6. Asegment = _____________
10
4
12
120˚
60˚
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)?
360 ft
160 ft
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.
(160π + 720) ≈ 1222.7 ft . 11. 2880 + 656π ≈ 4940.9ft2 . 14. A = (196 − 49π )u2 . 15. P = 14π u .
Goat on a Rope
Class Exercises
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?
Name: ___________________________
Date: ________________ Period: _____
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 AC
length AB
Ex.
D
E
0
1
F
2
3
4
5
G
H
6
7
I
8
9
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 B
area 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.
C
radius of circle B = 5 in.
radius of circle C = 10 in.
B
A
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.
Ex. Find the probability that a point chosen at random in this circle will be in the given
section.
E
(a) A
A
D
50°
(b) C
40°
60° 120°
(c) D
C
B
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
8 cm
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?
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?
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?
Review of Exponents, Distributing, Factoring
(As it relates to area on the TAKS test)
Name: ___________________________
Date: ________________ Period: _____
Exponents: Multiplying, Dividing, and Raising to a Power
3
* When in doubt, E X P A N D the exponent (x = x·x·x) and regroup or cancel pairs.
1. Which expression describes the area in a square
3 2
units of a rectangle that has a width of 4x y and
4 5
a length of 3x y ?
11 5
2. Which expression represents the area of a triangle
2 4
with a base of 2x y z units and a height of
4 3
5xy z units?
9 7
3. The area of a rectangle is 30m n square units.
4 2
If the length of the rectangle is 6m n units, how
many units wide is the rectangle?
(m ≠ 0 and n ≠ 0)
4. The area of a parallelogram is 40m n square
12 –3
units. If the base of the parallelogram is 5m n
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
6. Write an expression which represents the volume in
3
4
5x y ?
3
4
cubic units of a cube with a side length of 5x y .
(Formula for volume of a cube is V = s3)
a
b
Write the rule for multiplying two exponents with the same base (for example, x and x ).
a
b
Write the rule for dividing two exponents with the same base (for example, x and x ).
a
Write the rule for raising an exponent to a power (for example, raising x 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
12. Factor out the GCF: 12 x + 48 x − 36 x
4
2
factors: 4 x + 48 x − 36 x
3
2
Factoring to find the roots/solutions of equations like: ax + bx + c = 0
* Find factors of a·c that add to b.
2
2
13. Factor x + 5x + 6.
14. Factor x + 6x – 27.
2
2
15. Factor x – 17x + 60.
2
2
16. Factor x – x – 42.
2
17. Factor 2x + x – 6.
18. Factor 4x – 19x + 12.
19. The area of a rectangle is given by the equation
20. The area of a rectangle is 3 x + 14 x + 8 , and the
width is x + 4 . What expression best descibes the
rectangle’s length?
2l − 5l = 18 , in which l is the rectangle’s length.
2
What is the length of the rectangle?
2