Chapter 11 Resource Masters
NAME
DATE
11-1
PERIOD
Study Guide and Intervention
Areas of Parallelograms and Triangles
Areas of Parallelograms
Any side of a parallelogram can be called a base. The
height of a parallelogram is the perpendicular distance between any two parallel bases.
The area of a parallelogram is the product of the base and the height.
Example
Find the area of parallelogram EFGH.
B
h
D
E
Area of a parallelogram
A = bh
= 30(18)
b = 30, h = 18
= 540
Multiply.
The area is 540 square meters.
C
T b
Lesson 11-1
Area of a Parallelogram
A
If a parallelogram has an area of A square units,
a base of b units, and a height of h units, then
A = bh.
F
18 m
H
G
30 m
Exercises
Find the perimeter and area of each parallelogram. Round to the nearest tenth if
necessary.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
2.
3.
1.6 cm
16 ft
8 ft
4.
10 in.
60˚
1.6 cm
24 in.
10 ft
5.
8 in.
6.
15 ft
45°
30 ft
20 in.
4.1 cm
4.1 cm
7. TILE FLOOR A bathroom tile floor is made
of black-and-white parallelograms. Each
parallelogram is made of two triangles with
dimensions as shown. Find the perimeter
and area of one parallelogram.
Chapter 11
5
6.7
cm
7 cm
2 cm
15 cm
11 cm
Glencoe Geometry
NAME
DATE
11-1
PERIOD
Study Guide and Intervention (continued)
Areas of Parallelograms and Triangles
Areas Of Triangles The area of a triangle is one half the product of the base and its
corresponding height. Like a parallelogram, the base can be any side, and the height is the
length of an altitude drawn to a given base.
X
If a triangle has an area of A square units, a base of b
units, and a corresponding height of h units, then
1
A=−
bh.
Area of a Triangle
h
Z
2
Example
1
bh
A=−
Find the area of the triangle.
28 m
Area of a triangle
2
1
=−
(24)(28)
b = 24, h = 28
= 336
Multiply.
2
Y
b
24 m
The area is 336 square meters.
Exercises
Find the perimeter and area of each triangle. Round to the nearest tenth if
necessary.
1.
5 cm
3.
3 cm
15 in.
8 ft
4.
11 in.
34 ft
14 ft
40 in.
22 ft
5.
6.
9 cm
18 in.
13 cm
21 mm
9 cm
26 in.
12 in.
7. LOGO The logo for an engineering company is on
a poster at a job fair. The logo consists of two triangles
that have the dimensions shown. What are the
perimeter and area of each triangle?
21 mm
Triangle 2
Triangle 1
25 in.
20 in.
Chapter 11
6
25 in.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15 cm
2.
NAME
DATE
11-1
PERIOD
Skills Practice
Areas of Parallelograms and Triangles
Find the perimeter and area of each parallelogram or triangle. Round to the
nearest tenth if necessary.
1.
2.
5.5 ft
12 mm
4 ft
3.
60˚
10 mm
4.
14 yd
Lesson 11-1
18 mm
26 in.
22 in.
7 yd
5.
45˚
45˚
6.
3.4 m
18.5 km
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9 km
7.
60˚
30 cm
8.
20 cm
17 in.
13 in.
17 in.
9. The height of a parallelogram is 10 feet more than its base. If the area of the
parallelogram is 120 square feet, find its base and height.
10. The base of a triangle is one half of its height. If the area of the triangle is 196 square
millimeters, find its base and height.
Chapter 11
7
Glencoe Geometry
NAME
11-1
DATE
PERIOD
Practice
Areas of Parallelograms and Triangles
Find the perimeter and area of each parallelogram or triangle. Round to the
nearest tenth if necessary.
1.
2.
5m
11 m
60˚
4.
3.
8 cm
10 in.
45˚
45˚
10 cm
5.
6.
40 cm
17 cm
12.8 ft
15 cm
8 ft
20 in.
25 cm
12 in.
16 in.
4 ft
6 ft
7. The height of a parallelogram is 5 feet more than its base. If the area of the
parallelogram is 204 square feet, find its base and height.
9. The base of a triangle is four times its height. If the area of the triangle is
242 square millimeters, find its base and height.
10. FRAMING A rectangular poster measures 42 inches by 26 inches. A frame shop fitted
the poster with a half-inch mat border.
a. Find the area of the poster.
b. Find the area of the mat border.
c. Suppose the wall is marked where the poster will hang. The marked area includes
an additional 12-inch space around the poster and frame. Find the total wall area
that has been marked for the poster.
Chapter 11
8
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. The height of a parallelogram is three times its base. If the area of the parallelogram
is 972 square inches, find its base and height.
NAME
DATE
11-1
PERIOD
Word Problem Practice
Areas of Parallelograms and Triangles
1. PACKAGING A box with a square
opening is squashed into the rhombus
shown below.
4. PATHS A concrete path shown below is
made by joining several parallelograms.
102”
106”
14 in.
144”
48” 100”
48”
128”
What is the total area of the path?
What is the area of the opening?
2. RUNNING Jason jogs once around a city
block shaped like a parallelogram.
5. HIGHWAY SUPPORTS Three columns
are being placed at the vertices of a
right triangle to support a highway.
Two of the columns are marked on the
coordinate plane shown.
100 yd
y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5
200 yd
How far did Jason jog?
x
-5
O
5
a. What are two possible coordinates of
the third column to form a right
triangle?
3. SHADOWS A rectangular billboard
casts a shadow on the ground in the
shape of a parallelogram. What is the
area of the ground covered by the
shadow? Round your answer to the
nearest tenth.
b. What is the area in square units of
each of the two right triangles that
result from the possibilities you found
in Exercise a? Explain.
30 ft
15 ft
Chapter 11
9
Glencoe Geometry
Lesson 11-1
7 in.
NAME
11-1
DATE
PERIOD
Enrichment
Area of a Parallelogram
You can prove some interesting results using the formula
you have proved for the area of a parallelogram by drawing
auxiliary lines to form congruent regions. Consider the top
parallelogram shown at the right. In the figure, d is the
−−−
length of the diagonal BD, and k is the length of the
−−−
perpendicular segment from A to BD. Now consider the
second figure, which shows the same parallelogram with a
number of auxiliary perpendiculars added. Use what you
know about perpendicular lines, parallel lines, and
congruent triangles to answer the following.
A
B
k
d
D
C
A
B
k
F
E
H
k
1. What kind of figure is DBHG?
D
C d
G
2. If you moved △AFB to the lower-left end of figure
DBHG, would it fit perfectly on top of △DGC? Explain
your answer.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. Which two triangular pieces of #ABCD are congruent
to △CBH?
4. The area of #ABCD is the same as that of figure DBHG,
since the pieces of #ABCD can be rearranged to form
DBHG. Express the area of #ABCD in terms of the
measurements k and d.
Chapter 11
10
Glencoe Geometry
NAME
DATE
11-1
PERIOD
Graphing Calculator Activity
Cabri Junior: Areas of Parallelograms
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 1
Draw a parallelogram.
• Select F2 Segment to draw a segment.
• Select F5 Alph-num to label the endpoints of the segment A and B.
• Draw segment AD.
• Select F3 Parallel to draw a line parallel to segment AB through D. Select
point D, and then segment AB.
• Draw a line parallel to segment AD through B.
• Select F2 Point, Intersection to place a point at the intersection of the two
lines drawn. Label the point C.
• Select F2 Quad and draw a quadrilateral by selecting points A, B, C, and D.
Step 2
Find the measure of the area of parallelogram ABCD.
• Select F5 Measure, Area.
• Place the cursor on any segment of parallelogram ABCD.
Then press ENTER .
• The area appears with the hand attached. Move the
number to an appropriate place.
Step 3
Find the measure of the perimeter of parallelogram ABCD.
• Select F5 Measure, D. & Length.
• Place the cursor on any segment of parallelogram ABCD. Then press
• The area appears with the hand attached. Move the number to an
appropriate place.
ENTER
.
The perimeter of the parallelogram shown here is 16.2 units and the area is 13.8 square
units.
Exercises
Analyze your drawing.
1. Find the lengths of all four sides of the parallelogram.
2. Using the information from Exercise 1, what is the perimeter of the parallelogram? Does
this measurement match that found by Cabri Junior?
3. Construct a line segment showing the height of the parallelogram. What is the length of
the line segment?
4. What is the measure of the base of the parallelogram?
5. Using the information from Exercises 3 and 4, what is the area of the parallelogram?
Does this measurement match the one found by Cabri Junior?
6. Select one of the vertices and drag it to change the dimensions of the parallelogram.
(Press CLEAR so the pointer becomes a black arrow. Move the pointer close to a vertex
until the arrow becomes transparent and the vertex is blinking. Press ALPHA to change
the arrow to a hand. Then move the vertex.) Do you see any patterns or relationships?
Chapter 11
11
Glencoe Geometry
Lesson 11-1
Cabri Junior can be used to find the perimeters and areas of parallelograms.
NAME
11-1
DATE
PERIOD
Geometer’s Sketchpad Activity
Areas of Parallelograms
Exercises
Analyze your drawing.
1. Find the lengths of all four sides of the parallelogram.
2. Using the information from Exercise 1, what is the perimeter of the parallelogram? Does
this measurement match that found by the Geometer’s Sketchpad?
3. Construct a line segment showing the height of the parallelogram. What is the length of
the line segment?
4. What is the measure of the base of the parallelogram?
5. Using the information from Exercises 3 and 4, what is the area of the parallelogram? Does
this measurement match the one found by the Geometer’s Sketchpad?
6. Select one of the vertices and drag it to change the dimensions of the parallelogram. Do
you see any patterns or relationships?
Chapter 11
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The Geometer’s Sketchpad can be used to find the perimeters and areas
of parallelograms.
Step 1 Use The Geometer’s Sketchpad to draw a parallelogram.
• Construct a segment by selecting the Segment tool from the toolbar. First, click
the first point. Then click on a second point to draw the segment.
• Next, use one of the endpoints of the original segment as the first point for the
new segment and click on a second point to construct the new segment.
• Construct a parallel line to the original segment by first highlighting the
original segment and the endpoint not on that segment. Then select Parallel
Line from the Construct menu.
• Construct a parallel line to the second segment by highlighting the second
segment and the point not on it. Then select Parallel Line from the
Construct menu.
• Next, construct a point on the intersection of the two lines. Use the Point tool
from the toolbar to select the point where the two lines intersect.
• Construct the interior of the parallelogram by highlighting all four points and
selecting Quadrilateral Interior under the Construct menu.
Step 2 Use The Geometer’s Sketchpad to find
the perimeter of the parallelogram.
• Highlight the interior of the
Perimeter ABCD = 11.33 cm
parallelogram using the Selection
2
Area ABCD = 6.63 cm
Arrow tool from the toolbar.
• Next, find the perimeter by
B
A
selecting Perimeter under the
Measure menu.
C
D
Step 3 Use The Geometer’s Sketchpad to find
the area of the parallelogram.
• Highlight the interior of the
parallelogram using the Selection
Arrow tool from the toolbar.
• Next, find the area by selecting Area under the Measure menu.
The perimeter of the parallelogram shown here is 11.33 cm and the area is 6.63 cm2.
NAME
DATE
11-2
PERIOD
Study Guide and Intervention
Areas of Trapezoids, Rhombi, and Kites
Areas of Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel
sides, called bases. The height of a trapezoid is the perpendicular distance between the
bases. The area of a trapezoid is the product of one half the height and the sum of the
lengths of the bases.
Area of a Trapezoid
h
1
A=−
h (b1 + b2)
2
b2
Find the area of the trapezoid.
1
A=−
h(b1 + b2)
2
1
=−
(15)(18 + 40)
2
= 435
Area of a trapezoid
h = 15, b1 = 18, b2 = 40
18 m
Simplify.
15 m
The area of the trapezoid is 435 square meters.
40 m
Lesson 11-2
Example
b1
If a trapezoid has an area of A square units, bases of
b1 and b2 units, and a height of h units, then
Exercises
Find the area of each trapezoid.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
28 yd
2.
32 ft
16 ft
3.
32 m
24 yd
12 yd
20 m
18 ft
50 m
4.
5.
5 in.
38 cm
6.
30 ft
6 cm
21 cm
8 in.
16 ft
15 in.
18 ft
1
20 2 ft
7. OPEN ENDED Ryan runs a landscaping business.
A new customer has a trapezodial shaped backyard,
shown at the right. How many square feet of grass
will Ryan have to mow?
1
38 4 ft
5 ft
Chapter 11
13
5 ft
Glencoe Geometry
NAME
DATE
11-2
PERIOD
Study Guide and Intervention (continued)
Areas of Trapezoids, Rhombi, and Kites
Areas of Rhombi and Kites A rhombus is a parallelogram with all four sides
congruent. A kite is a quadrilateral with exactly two pairs of consecutive sides congruent.
If a rhombus or kite has an area of A square
units, and diagonals of d1 and d2 units, then
1
A=−
d1 · d2.
Area of Rhombus
or Kite
Example
d2
d2
d1
d1
2
Find the area of the rhombus.
1
A=−
d1d2
Area of rhombus
2
1
=−
(7)(9)
2
7 cm
d1 = 7, d2 = 9
= 31.5
Simplify.
m
9c
The area is 31.5 square meters.
Exercises
Find the area of each rhombus or kite.
1.
20 in.
26 in.
2.
3.
10 cm
cm
18 ft
24 ft
56 in. 20 in.
4.
1
m
3c
24 ft
10 ft
5.
6.
7m
8m
7m
12 cm
18 cm
ALGEBRA Find x.
7. A = 164 ft2
8. A = 340 cm2
x mm
12.8 ft
x ft
x cm
20 cm
20 ft
Chapter 11
9. A = 247.5 mm2
22.5 mm
14
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
20
NAME
DATE
11-2
PERIOD
Skills Practice
Areas of Trapezoids, Rhombi, and Kites
Find the area of each trapezoid, rhombus, or kite.
1.
2.
6m
12 mm
10 m
14 mm
15 m
4.
5 ft
11 in.
15 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
8 ft
7.5 in.
6.
4m
Lesson 11-2
3.
29 cm
23 cm
16 m
9.5 cm
ALGEBRA Find each missing length.
7. A trapezoid has base lengths of 6 and 15 centimeters with an area of 136.5 square
centimeters. What is the height of the trapezoid?
8. One diagonal of a kite is four times as long as the other diagonal. If the area of the kite
is 72 square meters, what are the lengths of the diagonals?
9. A trapezoid has a height of 24 meters, a base of 4 meters, and an area of 264 square
meters. What is the length of the other base?
Chapter 11
15
Glencoe Geometry
NAME
11-2
DATE
PERIOD
Practice
Areas of Trapezoids, Rhombi, and Kites
Find the area of each trapezoid, rhombus, or kite.
1.
2.
31 m
3.
5m
2.4 in.
34 cm
16.4 in.
16 m
11 cm
4.
6.5 ft
5.
6.
5 cm
17 ft
8 ft
2 cm
12 ft
21.5 ft
ALGEBRA Find each missing length.
7. A trapezoid has base lengths of 19.5 and 24.5 centimeters with an area of 154 cm2.
What is the height of the trapezoid?
9. A trapezoid has a height of 40 inches, a base of 15 inches, and an area of 2400 square
inches. What is the length of the other base?
10. DESIGN Mr. Hagarty used 16 congruent rhombi-shaped tiles to design
the midsection of the backsplash area above a kitchen sink. The length
of the design is 27 inches and the total area is 108 square inches.
a. Find the area of one rhombus.
b. Find the length of each diagonal.
Chapter 11
16
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. One diagonal of a kite is twice as long as the other diagonal. If the area of the kite is
400 square meters, what are the lengths of the diagonals?
NAME
11-2
DATE
PERIOD
Word Problem Practice
Areas of Trapezoids, Rhombi, and Kites
1. INTERIOR DESIGN The 20-by-20-foot
square shows an office floor plan
composed of three indoor gardens and
one walkway, all congruent in shape.
The gardens are centered around a
15-by-15 foot lounging area. What is
the area of one of these gardens?
4. HEXAGONS Heather makes a hexagon
by attaching two trapezoids together as
shown. What is the area of the hexagon?
20 cm
15 cm
30 cm
Garden
10 cm
Walkway
5. TILINGS Tile making often requires an
artist to find clever ways of dividing a
shape into several smaller, congruent
shapes. Consider the isosceles trapezoid
shown below.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. CUTOUTS A trapezoid is cut from a
6-inch-by-2-inch rectangle. The length of
one base is 6 inches. What is the area of
the trapezoid?
6 in.
45°
1 unit
2 in.
60˚
2
3
4
60˚
a. Show how to divide the trapezoid into
3 congruent triangles. What is the
area of each triangle?
3. SHARING Bernard has a birthday cake
shaped like a kite. He needs to cut it
into four pieces to share with three
friends. He divides the cake as shown
below. Which piece(s) is the largest?
What is the area of the cake?
1
2 units
1 unit
60˚
2 units
60˚
b. Show how to divide the trapezoid into
4 congruent trapezoids. What is the
area of each of the smaller
trapezoids?
10 in.
6 in.
1 unit
0.5
60˚
60˚
2 units
Chapter 11
17
Glencoe Geometry
Lesson 11-2
Garden
Garden
15 cm
NAME
DATE
11-2
PERIOD
Enrichment
Perimeters of Similar Figures
You have learned that if two figures are similar, the ratio of the lengths of the corresponding
sides are equal. If two figures are similar, then their perimeters are also proportional to the
scale factor between them.
Trapezoid II is k times larger than trapezoid I. Thus, its base is k times as large as that of
trapezoid I and its height its k times as large as that of trapezoid I.
ks2
side of trapezoid II
−− = −
s2 = k
side of trapezoid I
kb1
b1
s1
ks1
s2
h
kh
ks2
kb2
b2
k(s1 + s2 + b1 + b2)
perimeter trapezoid II
Trapezoid I
Trapezoid II
−− = −−
= k Perimeter = s + s + b + b Perimeter = ks + ks + kb + kb
1
2
1
2
1
2
1
2
perimeter trapezoid I
s1 + s2 + b1 + b2
= k (s1 + s2 + b1 + b2)
Solve.
−− −−
1. Trapezoid ABCD ~ trapezoid EFGH.
2. In the figure, EF ∥ AB and the
EF = 10, GH = 8, HE = GF = 5, and AB = 5.
perimeter of trapezoid ABCD is 56.
Find the perimeter of trapezoid ABCD.
Find the perimeter of trapezoid EFCD.
Round to the nearest tenth.
"
)
$
# &
%
(
&
'
"
$
12
18
'
#
3. Two similar trapezoids have perimeters of 37.5 feet and 150 feet. The length of a side of
the smaller trapezoid is 10 feet. Find the length of the corresponding side of the larger
trapezoid.
4. Find the ratio of the perimeters of two similar trapezoids if the lengths of two
corresponding sides of the trapezoids are 9 centimeters and 27 centimeters.
Chapter 11
18
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
%
NAME
11-3
DATE
PERIOD
Study Guide and Intervention
Areas of Circles and Sectors
Areas Of Circles
If a circle has an area of A square units
and a radius of r units, then A = πr2.
Area of a Circle
Example
The area of a circle is equal to π times the square of radius.
O
r
Find the area of the circle p.
A = πr2
Area of a circle
= π(6) 2
r=6
≈ 113.04
Use a calculator.
12 m
p
The area of the circle is about 113.04 square meters.
If d = 12m, then r = 6m.
Exercises
Find the area of each circle. Round to the nearest tenth.
2.
20 m
9.5
Lesson 11-3
in.
5 in.
3.
4.
5.
6.
11 ft
m
11 in.
88
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
Find the indicated measure. Round to the nearest tenth.
7. The area of a circle is 153.9 square centimeters. Find the diameter.
8. Find the diameter of a circle with an area of 490.9 square millimeters.
9. The area of a circle is 907.9 square inches. Find the radius.
10. Find the radius of a circle with an area of 63.6 square feet.
Chapter 11
19
Glencoe Geometry
NAME
DATE
11-3
PERIOD
Study Guide and Intervention (continued)
Areas of Circles and Sectors
Areas of Sectors
intercepted arc.
A sector of a circle is a region bounded by a central angle and its
If a sector of a circle has an area of A square units,
a central angle measuring x°, and a radius of r units,
x
then A = −
πr2.
Area of a Sector
360
Example
Find the area of the shaded sector.
x
A=−
# πr2
360
36
=−
# π(5)2
360
≈ 7.85
Area of a sector
36° 5 in.
x = 36 and r = 5
Use a calculator.
The area of the sector is about 7.85 square inches.
Exercises
Find the area of each shaded sector. Round to the nearest tenth.
1.
2.
3
ft
3.
100°
10
% '
20°
m
7m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
#
+
,
45°
&
"
$
"
$
15°
,
88°
36°
10 m
'
-
+
7. SANDWICHES For a party, Samantha wants to
have finger sandwiches. She cuts sandwiches into
circles. If she cuts each circle into three congruent
pieces, what is the area of each piece?
Chapter 11
&
cm
#
20 ft
6. %
7.
5
5.
4.
20
2.5 in.
Glencoe Geometry
NAME
DATE
11-3
PERIOD
Skills Practice
Areas of Circles and Sectors
Find the area of each circle.
1.
2.
3.
10.5 m
7m
18 in.
Find the indicated measure. Round to the nearest tenth.
4. The area of a circle is 132.7 square centimeters. Find the diameter.
5. Find the diameter of a circle with an area of 1134.1 square millimeters.
7. Find the radius of a circle with an area of 2827.4 square feet.
Lesson 11-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. The area of a circle is 706.9 square inches. Find the radius.
Find the area of each shaded sector. Round to the nearest tenth.
8.
"
$
51°
9.
+
10.
%
2m
12.5 m
#
130°
,
'
117°
&
18 m
11. GAMES Jason wants to make a spinner for a new board game
he invented. The spinner is a circle divided into 8 congruent
pieces, what is the area of each piece to the nearest tenth?
16 cm
Chapter 11
21
Glencoe Geometry
NAME
DATE
11-3
PERIOD
Practice
Areas of Circles and Sectors
Find the area of each circle. Round to the nearest tenth.
1.
2.
24
in.
1.5 m
3.
4.5 cm
Find the indicated measure. Round to the nearest tenth.
4. The area of a circle is 3.14 square centimeters. Find the diameter.
5. Find the diameter of a circle with an area of 855.3 square millimeters.
6. The area of a circle is 201.1 square inches. Find the radius.
7. Find the radius of a circle with an area of 2290.2 square feet.
Find the area of each shaded sector. Round to the nearest tenth.
"
$
19 m
9.
37°
10.
&
6 in
%
'
#
,
8°
+
10
cm
128°
-
11. CLOCK Sadie wants to draw a clock face on a circular piece of cardboard. If the clock
face has a diameter of 20 centimeters and is divided into congruent pieces so that each
sector is 30°, what is the area of each piece?
Chapter 11
22
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8.
NAME
11-3
DATE
PERIOD
Word Problem Practice
Areas of Circles and Sectors
1. LOBBY The lobby of a bank features a
large marble circular table. The
diameter of the circle is 15 feet.
4. SOUP CAN Julie needs to cover the top
and bottom of a can of soup with
construction paper to include in her art
project. Each circle has a diameter of
7.5 centimeters. What is the total area
of the can that Julie must cover?
15 ft
What is the area of the circular table?
Round your answer to the nearest
tenth.
5. POOL A circular pool is surrounded by
a circular sidewalk. The circular
sidewalk is 3 feet wide. The diameter of
the sidewalk and pool is 26 feet.
Pool
Sidewalk
Diameter of
sidewalk and
pool = 26 ft
a. What is the diameter of the pool?
3. PEACE SYMBOL The symbol below, a
circle separated into 3 equal sectors, has
come to symbolize peace.
b. What is the area of the sidewalk
and pool?
r
c. What is the area of the pool?
Suppose the circle has radius r. What is
the area of each sector?
Chapter 11
23
Glencoe Geometry
Lesson 11-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. PORTHOLES A circular window on a
ship has a radius of 8 inches. What is
the area of the window? Round your
answer to the nearest hundredth.
NAME
DATE
11-3
PERIOD
Enrichment
Perimeter of a Sector
You have learned how to find the area of a sector of a circle using a ratio of the circle and
the area formula. Now you will learn how to find the perimeter of the sector of the circle.
The perimeter of the sector is the sum of the lengths
of two radii and the length of its arc.
!
= 2r + length of AB
P
"
6 in.
100°
sector
$
!.
Step 1 Find the length of AB
The length of the arc is a section of
the circumference. Multiply the ratio
of the degree measure of the
intercepted arc to 360° by the
circumference of the circle.
x
Length of arc = −
# 2(π)(r)
360
100
! = − # 2(π)(6)
Length of AB
360
#
x = 100 and r = 6
≈ 10.5
Use a calculator.
Step 2 Use the formula for the perimeter of a sector.
!
Psector = 2r + length of AB
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
≈ 2(6) + 10.5
≈ 22.5
The perimeter of the sector is about 22.5 inches.
Exercises
Find the perimeter of the shaded sector. Round to the nearest tenth.
1.
3
2.
50°
,
5 ft
.
3.
&
5
4.
15 cm
Chapter 11
150°
48°
7
1
'
170°
:
9
8m
12.5 m
.
(
24
Glencoe Geometry
NAME
DATE
11-4
PERIOD
Study Guide and Intervention
Areas of Regular Polygons and Composite Figures
Areas of Regular Polygons
In a regular polygon, the segment drawn from the center
of the polygon perpendicular to the opposite side is called the apothem. In the figure at the
−−
−−−
right, AP is the apothem and AR is the radius of the circumscribed circle.
U
Area of a Regular Polygon
If a regular polygon has an area of A square units,
a perimeter of P units, and an apothem of a units,
1
then A = −
aP.
V
2
R
Example 1
Verify the formula
1
A = − aP for the regular pentagon above.
2
For ∠RAS, the area is
2
(2)
1
(RS)(AP). Substituting
pentagon is A = 5 −
P for 5RS and substituting a for AP, then
S
Example 2
Find the area of regular
pentagon RSTUV above if its perimeter
is 60 centimeters.
RP
tan m∠RAP = −
1
aP.
A=−
AP
6
tan 36 = −
AP
6
AP = −
tan 36
2
≈ 8.26
1
1( )
aP = −
60 (8.26) or 247.8.
So, A = −
2
2
The area is about 248 square centimeters.
Exercises
Find the area of each regular polygon. Round to the nearest tenth.
1.
2.
14 m
3.
15 in.
10 in.
4.
5.
6.
5√3 cm
10.9 m
7.5 m
10 in.
Chapter 11
Lesson 11-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
P
First find the apothem.
360
or
The measure of central angle RAS is −
5
72. Therefore, m∠RAP = 36. The perimeter
is 60, so RS = 12 and RP = 6.
1
1
bh = −
(RS)(AP). So the area of the
A=−
2
T
A
25
Glencoe Geometry
NAME
DATE
11-4
PERIOD
Study Guide and Intervention (continued)
Areas of Regular Polygons and Composite Figures
Areas of Composite Figures
A composite figure is a figure that can be seprated into
regions that are basic figures. To find the area of a composite figure, separate the figure
into basic figures of which we can find the area. The sum of the areas of the basic figures is
the area of the figure.
Example
a.
Find the area of the shaded region.
b.
50 ft
5 cm
30 ft
The figure is a rectangle minus one half of
a circle. The radius of the circle is one half
of 30 or 15.
1
πr2
A = lw - −
2
= 50(30) - 0.5π(15)2
≈ 1146.6 or about 1147 ft2
The dimensions of the rectangle are
10 centimeters and 30 centimeters. The area
of the shaded region is
(10)(30) - 3π(52) = 300 - 75π
≈ 64.4 cm2
Exercises
Find the area of each figure. Round to the nearest tenth if necessary.
34 ft
1.
2.
3.
14 cm
38 cm
24 in.
24 in.
10 cm
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15 ft
40 in.
22 cm
40 cm
42 cm
5.
6.
64 m
20 m
Chapter 11
40 m
35 yd
15 yd
20 m
26
Glencoe Geometry
NAME
DATE
11-4
PERIOD
Skills Practice
Areas of Regular Polygons and Composite Figures
Find the area of each regular polygon. Round to the nearest tenth.
1.
2.
10 cm
8m
3.
4.
6 ft
Find the area of each figure. Round to the nearest tenth if necessary.
5.
6.
5m
3 ft
12 m
7 ft
20 m
7.
8.
15 cm
8 in.
30 cm
8 in.
Chapter 11
27
Glencoe Geometry
Lesson 11-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15 in.
NAME
DATE
11-4
PERIOD
Practice
Areas of Regular Polygons and Composite Figures
Find the area of each regular polygon. Round to the nearest tenth.
1.
2.
14 cm
7m
Find the area of each figure. Round to the nearest tenth if necessary.
3.
4.
20 mm
38 ft
22 ft
20 mm
22 ft
5.
6.
9m
20 in.
30 in.
13 in.
7m
23 m
7. LANDSCAPING One of the displays at a botanical garden is a koi pond with a walkway
around it. The figure shows the dimensions of the pond and the walkway.
7 ft
15 ft
13 ft
35 ft
a. Find the area of the pond to the nearest tenth.
b. Find the area of the walkway to the nearest tenth.
Chapter 11
28
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13 in.
NAME
DATE
11-4
PERIOD
Word Problem Practice
Areas of Regular Polygons and Composite Figures
1. YIN-YANG SYMBOL A well-known
symbol from Chinese culture is the yinyang symbol, shown below.
4. TRACK A running track has an inner
and outer edge. Both the inner and outer
edges consist of two semicircles joined by
two straight line segments. The straight
line segments are 100 yards long.
The radii of the inner
edge semicircles are
100 yd
25 yd
25 yards each and the
radii of the outer edge
semicircles are 32 yards each. What is
the area of the track? Round your
answer to the nearest hundredth of
a yard.
Suppose the large circle has radius r,
r
the small circles have radius −
, and the
8
S-curve is two semicircles, each with
r
radius −
. In terms of r, what is the area
5. SEMICIRCLES Bridget arranged three
semicircles in the pattern shown.
2
2. PYRAMIDS Martha’s clubhouse is
shaped like a square pyramid with four
congruent equilateral triangles for its
sides. All of the edges are 6 feet long.
What is the total surface area of the
clubhouse including the floor? Round
your answer to the nearest hundredth.
The right triangle has side lengths 6, 8,
and 10 inches.
a. What is the total area of the three
semicircles? Round your answer to
the nearest hundredth of a square
inch.
3. MINIATURE GOLF The plan for a
miniature golf hole is shown below. The
right angle in the drawing is a central
angle.
b. If the right triangle had side lengths
√##
##, and 10 inches, what would
21 , √79
the total area of the three semicircles
be? Round your answer to the nearest
hundredth of a square inch.
1.7 m
3m
What is the area of the playing surface?
Round your answer to the nearest
hundredth of a square meter.
Chapter 11
29
Glencoe Geometry
Lesson 11-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
of the black region?
NAME
DATE
11-4
PERIOD
Enrichment
Areas of Inscribed Polygons
A protractor can be used to inscribe a regular polygon in a circle.
Follow the steps below to inscribe a regular nonagon in ⊙N.
Step 1 Find the degree measure of each of
the nine congruent arcs.
Step 2 Draw 9 radii to form 9 angles with
the measure you found in Step 1.
The radii will intersect the circle in
9 points.
Step 3 Connect the nine points to form the
N
nonagon.
1. Find the length of one side of the
nonagon to the nearest tenth of a
centimeter. What is the perimeter of
the nonagon?
2. Measure the distance from the center
perpendicular to one of the sides of the nonagon.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. What is the area of one of the nine triangles formed?
4. What is the area of the nonagon?
Make the appropriate changes in Steps 1–3 above to inscribe
a regular pentagon in ⊙P. Answer each of the following.
5. Use a protractor to inscribe a
regular pentagon in ⊙P.
6. What is the measure of each of
the five congruent arcs?
7. What is the perimeter of the
pentagon to the nearest tenth
of a centimeter?
P
8. What is the area of the pentagon
to the nearest tenth of a
centimeter?
Chapter 11
30
Glencoe Geometry
NAME
DATE
11-5
PERIOD
Study Guide and Intervention
Areas of Similar Figures
Areas of Similar Figures If two polygons are similar, then their areas are
proportional to the square of the scale factor between them.
Example
△JKL ∼ △PQR.
The area of △JKL is 40 square inches.
Find the area of △PQR.
+
6
12
Find the scale factor: −
or −
.
10
10 in.
5
6 2
The ratio of their areas is − .
5
area of △PQR
6
− = −
area of△JKL
(5)
3
()
,
2
Write a proportion.
area of △PQR
36
− =−
(5)
6
Area of △JKL = 40; −
40
25
36
area of △PQR = −
# 40
25
2
-
1
12 in.
2
36
=−
25
Multiply each side by 40.
area of △PQR = 57.6
Simplify.
So the area of △PQR is 57.6 square inches.
Exercises
1.
2.
5 m
15 m
2 in.
6 in.
A = 20 in2
15.5 cm
4.
A = 200 cm2
t
10.5 cm
16
f
3.
t
A = 12 m2
20
f
A = 8050 ft2
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For each pair of similar figures, find the area of the shaded figure.
Chapter 11
31
Glencoe Geometry
NAME
DATE
11-5
PERIOD
Study Guide and Intervention (continued)
Areas of Similar Figures
Scale Factors and Missing Measures in Similar Figures
You can use the areas
of similar figures to find the scale factor between them or a missing measure.
Example
"
If !ABDC is similar to
!FGJH, find the value of x.
Let k be the scale factor between !ABDC and !FGJH.
Theorem 11.1
64
−
= k2
Substitution
8
−
=k
Take the positive square root of each side.
49
7
$
% )
10 m
A = 64 m2
area !ABCD
−
= k2
area !FGJH
# '
(
x
+
A = 49 m2
Use this scale factor to find the value of x.
CD
−
=k
The ratio of corresponding lengths of similar polygons is equal to the scale factor between the
HJ
8
x
−
=−
10
7
8
x = − # 10 or 11.4
7
polygons.
Substitution
Multiply each side by 10.
For each pair of similar figures, use the given areas to find the scale factor from
the unshaded to the shaded figure. Then find x.
1.
2.
x ft
8
x
in
.
28 ft
A = 296 ft2
A = 54 in2
A = 216
A = 169 ft2
in2
3.
4.
7 ft
x cm
A = 300 cm2
Chapter 11
x
21 cm
A = 900 cm2
A = 50 ft2
32
A = 30 ft2
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
11-5
PERIOD
Skills Practice
Areas of Similar Figures
For each pair of similar figures, find the area of the shaded figure.
1.
2.
2 in.
8.5 in.
44 m
11 m
A = 20
A = 34 in2
m2
For each pair of similar figures, use the given areas to find the scale factor from
the unshaded to the shaded figure. Then find x.
3.
4.
21 m
12 ft
x
x
A = 10 ft2
A = 360 ft2
A = 510 m2
5.
6.
9.5 in.
x
A = 16 in2
A = 71 in2
14 ft
x
A = 588 ft2
A = 272 ft2
7. SCIENCE PROJECT Matt has two posters for his science project. Each poster is a
rectangle. The length of the larger poster is 11 inches. The length of the smaller
poster is 6 inches. What is the area of the smaller poster if the larger poster is
93.5 square inches?
Chapter 11
33
Glencoe Geometry
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A = 4590 m2
NAME
DATE
11-5
PERIOD
Practice
Areas of Similar Figures
For each pair of similar figures, find the area of the shaded figure.
1.
2.
16 m
3m
20 in.
A = 38 m2
30 in.
A = 200 in2
For each pair of similar figures, use the given areas to find the scale factor from
the unshaded to the shaded figure. Then find x.
3.
4.
x cm
7 cm
8m
xm
A = 50 m2
A = 30 cm2
A = 72 m2
A = 70 cm2
x ft
6.
8 ft
9 cm
A = 39 cm2
A = 16 ft2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
x cm
A = 13 cm2
A = 64 ft2
7. ARCHERY A target consists of two concentric similar
octagons. The outside octagon has a side length of 2 feet
and an area of 19.28 square feet. If the inside octagon has
a side length of 1.5 feet, what is the area of the inside octagon?
Chapter 11
34
Glencoe Geometry
NAME
11-5
DATE
PERIOD
Word Problem Practice
Areas of Similar Figures
1. CHANGING DIMENSIONS A polygon
has an area of 225 square meters. If the
area is tripled, how does each side length
change?
4. FOUNTAIN A local park has two
fountains in the shape of similar
trapezoids as shown.
Large
Fountain
100 ft.
2. CAKE Smith’s Bakery is baking several
large cakes for a community festival. The
cakes consist of two geometrically similar
shapes as shown. If 50 pieces of cake can
be cut from the smaller cake, how many
pieces of the same size can be cut from
the larger cake? Round to the nearest
piece of cake.
Large
cake
5 ft
Small
Fountain
40 ft.
A cement company charges $1000 to
pour the cement needed to go under the
smaller fountain. How much should the
town budget for the cement for both
fountains? Explain.
Smaller
cake
2 ft
1.6 ft
5. SCULTPURE An artist creates metal
sculptures in the shape of regular
octagons. The side length of the larger
sculpture is 7 inches, and the area of
the base of the smaller sculpture is
19.28 square inches.
3. PINS Carla has a shirt with decorative
pins in the shape of equilateral triangles.
The pins come in two sizes. The larger
pin has a side length that is three times
longer than the smaller pin. If the area
of the smaller pin is 6.9 square
centimeters, what is the approximate
area of the larger pin?
a. What is the side length of the smaller
sculpture?
b. The artist is going to pack the
sculptures in a circular box to take them
to an art show. Will the larger sculpture
fit in a circular box with a 15-inch
diameter? Explain your reasoning.
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 ft
Chapter 11
35
Glencoe Geometry
NAME
DATE
11-5
PERIOD
Enrichment
Areas of Similar Figures You have learned that to find the area of a composite figure,
you find the area of each basic figure and then use the Area Addition Postulate. You have
also learned that if two figures are similar, then their areas are proportional to the square
of the scale factor between them.
You can find the area of similar composite figures using this knowledge.
10 ft
Find the area of composite figure B.
6 ft
Step 1: Find the area of composite figure A.
3 ft
Area of rectangle = (3 ft)(7 ft) = 21 ft2
1
Area of trapezoid = −
(6 ft)(10 ft + 3 ft) = 39 ft2
2
7 ft
4 ft
Area of composite figure A = 21 ft + 39 ft = 60 ft .
2
2
2
3 ft
Step 2: Use scale factor to find the area of composite figure B.
#
"
area composite figure A
7 2
−− = −
4
area composite figure B
49
=−
16
60 ft2
49
−− = −
16
area composite figure B
16
2
area composite figure B = 60 " −
49 = 19.6 ft
()
So the area of composite figure B is about 19.6 square feet.
2. Jim is making a scale model of his
rectangular backyard and circular pool.
If the scale factor is 1:20, what is the
area of his model?
20 ft
12 in.
15 ft
10 ft
6 in.
10 in.
14 in.
real backyard
21 in.
"
real pool
#
4. Composite figure A and composite
figure B are similar. The length of the
sides of composite figure A is two-thirds
the length of the sides of composite
figure B. If the area of composite figure A
is 240 cm2, find the area of composite
figure B.
3. Composite figure A is similar to
composite figure B. Find the value of x
in composite figure B.
#
"
x
2 ft
A = 124 ft2
A = 1116 ft2
15 cm
22.5 cm
A = 240 cm2
Chapter 11
36
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve.
1. Composite figure A is
similar to composite
figure B. Find the area of
composite figure B.