Area
© 2011 Carnegie Learning
Sleds,
diamonds, and
deltas. Those are the
most popular kinds of kites.
But there are also rollers,
dragons, doperos, and
rokkakus. Kite flying is a
popular activity around
the world.
13.1 The Language of Geometry
Sketching, Drawing, Naming, and Sorting Basic Geometric Figures.............................................. 819
13.2 Weaving a rug
Area and Perimeter of Rectangles and Squares............833
13.3 Boundary Lines
Area of Parallelograms and Triangles. ........................ 847
13.4 The Keystone Effect
Area of Trapezoids. .................................................... 863
13.5 Go Fly a Kite
Area of Rhombi and Kites............................................ 875
13.6 Street Signs
Area of Regular Polygons........................................... 883
817
© 2011 Carnegie Learning
818 • Chapter 13 Area
The Language of Geometry
Sketching, Drawing, Naming, and
Sorting Basic Geometric Figures
Learning Goals
Key Terms
In this lesson, you will:








 Classify geometric figures
as polygons, triangles,
quadrilaterals, pentagons,
hexagons, heptagons,
octagons, nonagons, and
decagons.
 Define consecutive sides
and opposite sides.
protractor
straightedge
sketch
 right triangle
 obtuse
triangle
draw
construct
triangle
equilateral
 isosceles
categories.
triangle
compass
triangle
 Sort polygons into
 acute
triangle
 scalene
triangle
 equiangular
triangle
 quadrilateral
 opposite
 trapezoid
 isosceles
trapezoid
 polygon
 regular
polygon
 irregular
sides
 consecutive
sides





square
rectangle
rhombus
parallelogram
polygon






pentagon
hexagon
heptagon
octagon
nonagon
decagon
kite
W
hen you sketch a geometric figure, you create the figure without the use
© 2011 Carnegie Learning
of tools. When you draw a geometric figure, you create the figure with the use of
tools. A drawing is more accurate than a sketch. Any tools can be used, such as
rulers, straightedges, compasses, protractors, etc.
Look at the figure shown.
Make a sketch of the figure, and then make a drawing of
the figure. Compare your sketches and drawings to your
classmates. What do you notice? Were the sketches or
drawings more exact copies of the figure shown?
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 819
Problem 1 Triangles
Producing pictures, sketches, diagrams, and drawings of figures is a very important part of
geometry. Many tools can be used to create geometric figures. Some tools, such as a ruler
or a protractor, are classified as measuring tools. A protractor can be used to approximate
the measure of an angle. A compass is a tool used to create arcs and circles. A
straightedge is a ruler with no numbers. It is important to know when to use each tool.
●
When you sketch a geometric figure, the figure is created without the use of tools.
●
When you draw a geometric figure, the figure is created with the use of tools such as a
ruler, a straightedge, a compass, or a protractor. A drawing is generally more accurate
than a sketch.
●
When you construct a geometric figure, the figure is created using only a compass and
a straightedge.
You have already worked with basic geometric figures such as points, lines, rays, planes,
line segments, and angles. These basic figures can be used to build more complex
geometric figures. A triangle is the simplest closed three-sided geometric figure.
1. Sketch three different triangles.
A triangle is named using three capital letters representing the vertices, listed in a clockwise
or counterclockwise order. Triangle RAD can be written using symbols as nRAD. This is
read as “triangle RAD.” The triangle shown could be named nRAD, nADR, or nDRA.
A
D
R
The root word “tri” means “three,” so triangle literally means “three angles.” A triangle is a
closed figure because it has a well-defined interior and exterior.
820 • Chapter 13 Area
© 2011 Carnegie Learning
2. How many line segments, angles, and vertices are needed to form a triangle?
3. Label the vertices of the triangles you sketched in Question 1, and then use symbols
to name each triangle.
4. Name the three sides and the three angles of nRAD.
Sides:
Angles:
Don't
forget the
symbols when
naming sides, like
AD, and angles,
like DAR.
Triangles are classified by their side lengths or by their angle measures.
The root word “equi” means “equal,” and the root word “lateral” means
“side.” An equilateral triangle is a triangle with all sides congruent.
5. Draw an equilateral triangle. Label the drawing and include
measurements that verify it is an equilateral triangle.
© 2011 Carnegie Learning
An isosceles triangle is a triangle with at least two congruent sides.
So, an
equilateral
triangle is
also an
isosceles
triangle.”
6. Draw an isosceles triangle. Label the drawing and include measurements that verify it
is an isosceles triangle.
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 821
A scalene triangle is a triangle with no congruent sides.
7. Draw a scalene triangle. Label the drawing and include measurements that verify it is
a scalene triangle.
An equiangular triangle is a triangle with all angles congruent.
8. Draw an equiangular triangle. Label the drawing and include measurements that verify
it is an equiangular triangle.
9. How are equilateral and equiangular triangles related?
An acute triangle is a triangle that has three angles that each
measure less than 90°.
10. Draw an acute triangle. Label the drawing and include
measurements that verify it is an acute triangle.
822 • Chapter 13 Area
© 2011 Carnegie Learning
Time to get
out your
protractor.
A right triangle is a triangle that has a right angle.
11. Draw a right triangle. Label the drawing and include measurements that verify it is a
right triangle.
An obtuse triangle is a triangle that has an angle measuring greater than 90°.
12. Draw an obtuse triangle. Label the drawing and include measurements that verify it is
an obtuse triangle.
Problem 2 Quadrilaterals
A quadrilateral is a closed four-sided geometric figure.
Opposite sides are sides that do not share a common endpoint.
© 2011 Carnegie Learning
Consecutive sides are sides that share a common endpoint.
1. Sketch three different quadrilaterals.
2. How many line segments, angles, and vertices are needed to form a quadrilateral?
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 823
A quadrilateral is named using four capital letters representing the vertices, listed in a
clockwise or counterclockwise order. The quadrilateral shown could be named
quadrilateral ABCD, quadrilateral ADCB, quadrilateral BCDA, and so on.
A
B
D
C
3. What are three other names for quadrilateral ABCD not listed above?
4. Name the two pairs of opposite sides of quadrilateral ABCD.
5. Name the four pairs of consecutive sides of quadrilateral ABCD.
Sides:
Angles:
The root word “quad” means “four,” and the root word “lateral” means “side,” so
quadrilateral literally means “four sides.” A quadrilateral is a closed figure because it has a
well-defined interior and exterior.
7. Label the vertices of the quadrilaterals you sketched in Question 1, and then name
each quadrilateral.
824 • Chapter 13 Area
© 2011 Carnegie Learning
6. Name the four sides and the four angles of quadrilateral ABCD.
A square is a quadrilateral with all sides congruent and all angles congruent.
8. Draw a square. Label the drawing and include measurements.
A rectangle is a quadrilateral with opposite sides congruent and all angles congruent.
9. Draw a rectangle that is not a square. Label the drawing and include measurements.
So, are
all squares also
rectangles? Or, are all
rectangles also
squares?
A rhombus is a quadrilateral with all sides congruent. The plural of
rhombus is rhombi.
10. Draw a rhombus that is not a square. Label the drawing and
© 2011 Carnegie Learning
include measurements.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
11. Draw a parallelogram that is not a rectangle or a rhombus. Label the drawing and
include measurements.
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 825
A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite sides
that are not congruent.
12. Draw a kite. Label the drawing and include measurements.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
13. Draw a trapezoid. Label the drawing and include measurements.
An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent.
Problem 3 Polygons
Triangles and quadrilaterals are examples of geometric figures with many sides. The root
word “poly” means “many” and the root word “gon” means “side.” A polygon is a closed
figure that is formed by joining three or more line segments at their endpoints. A regular
polygon is a polygon with all sides congruent and all angles congruent. An irregular
polygon is a polygon that is not regular. Polygons are named using capital letters
representing the vertices, listed in a clockwise or counterclockwise order.
826 • Chapter 13 Area
© 2011 Carnegie Learning
14. Draw an isosceles trapezoid. Label the drawing and include measurements.
The root word “penta” means “five,” so pentagon literally means “five sides.” A pentagon
is a five-sided polygon.
1. Sketch a pentagon. Label the vertices and name the pentagon.
The root word “hexa” means “six,” so hexagon literally means “six sides.” A hexagon is a
six-sided polygon.
2. Sketch a hexagon. Label the vertices and name the hexagon.
The root word “hepta” means “seven,” so heptagon literally means “seven sides.” A
heptagon is a seven-sided polygon.
© 2011 Carnegie Learning
3. Sketch a heptagon. Label the vertices and name the heptagon.
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 827
The root word “octa” means “eight,” so octagon literally means “eight sides.” An octagon
is an eight-sided polygon.
4. Sketch an octagon. Label the vertices and name the octagon.
The root word “nona” means “nine,” so nonagon literally means “nine sides.” A nonagon
is a nine-sided polygon.
5. Sketch a nonagon. Label the vertices and name the nonagon.
The root word “deca” means “ten,” so decagon literally means “ten sides.” A decagon is
a ten-sided polygon.
© 2011 Carnegie Learning
6. Sketch a decagon. Label the vertices and name the decagon.
828 • Chapter 13 Area
Problem 4 Sort Activity
1. Cut out each shape shown in the diagram. How would you sort these shapes?
A
B
C
D
F
G
E
I
K
J
H
M
P
N
R
L
O
Q
© 2011 Carnegie Learning
U
W
T
S
V
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 829
© 2011 Carnegie Learning
830 • Chapter 13 Area
2. Tyler sorted all of the figures shown into two groups. One group contained all of the
figures with at least one pair of perpendicular sides. The second group contained all
of the figures that did not have at least one pair of perpendicular sides. Which figures
were in Tyler’s groups?
Group 1
Group 2
Figures that have at least one pair of
Figures that do not have at least one
perpendicular sides:
pair of perpendicular sides:
3. Molly sorted all of the figures shown into two groups. What characteristics describe
each of her two groups?
Group 1
Group 2
A, E, F, G, H, I, J, K, L, M, N, O, P, T,
B, C, D, Q, R, S, U
V, W
4. Carefully analyze the cut out figures. Think of a few characteristics different from the
ones Tyler and Molly used in the last two questions. Sort the figures into groups using
these characteristics. Explain your reasoning.
© 2011 Carnegie Learning
How
are the figures
in each of your
groups similar?
5. Compare the characteristics you used to those of your
classmates. What different characteristics did your
classmates use to sort the figures?
Be prepared to share your solutions and methods.
13.1 Sketching, Drawing, Naming, and Sorting Basic Geometric Figures • 831
© 2011 Carnegie Learning
832 • Chapter 13 Area
Weaving a Rug
Area and Perimeter of
Rectangles and Squares
Learning Goals
In this lesson, you will:





Calculate the area of rectangles and squares.
Calculate the perimeter of rectangles and squares.
Write a formula for the perimeter and area of a rectangle and a square.
Determine the effect of altering the dimensions of a rectangle or a square on the perimeter and area.
Calculate the area of composite figures.
C
arpets and rugs have been art forms in the Middle East for centuries. Its
history spans before the religions of Islam and Christianity. What is even more
remarkable is that carpet making and rug weaving is still an important part of the economy of many Middle Eastern countries. Many experts have estimated
that rug weaving brought in approximately $420 million dollars to the Iranian
economy in 2008. And rug weaving is quite a popular occupation. It is estimated
that there are 1.2 million rug weavers in Iran. With rugs and carpets still being
made by hand, do you think that carpets and rugs can be made by machine? Do
© 2011 Carnegie Learning
you think machine made Persian carpets and rugs will have the same quality of
handmade rugs?
13.2 Area and Perimeter of Rectangles and Squares • 833
Problem 1 A Rectangular Rug
Tyson is currently creating rectangular-shaped rugs.
1. One rectangular rug is seven feet long and three feet wide. Draw a model of this rug on
the grid shown. Each square on the grid represents a square that is one foot long and
one foot wide.
2. What is the area of this rug? Explain your calculation.
3. What is the perimeter of this rug? Explain your calculation.
834 • Chapter 13 Area
© 2011 Carnegie Learning
Are the
units of measure
the same for area
and perimeter?
4. Six different rectangles are drawn on the grid shown. The letters A through F name
each rectangle.
C
A
B
E
F
D
5. Each square on the grid represents a square that is one foot long and one foot wide.
Complete the table to show the length, width, area, and perimeter of each rectangle.
Rectangle
Length (units)
Width (units)
Perimeter (units)
Area (square units)
A
B
C
D
© 2011 Carnegie Learning
E
F
13.2 Area and Perimeter of Rectangles and Squares • 835
6. What is an example of two rectangles having the same area, but different dimensions?
a. What are the perimeters of these rectangles?
b. If the areas are equal, are the perimeters always equal?
7. You can determine the perimeter of a rectangle without drawing it if you know the
rectangle’s length and width. Explain how you can do this. Use the table in Question 5
to help you.
8. Write a formula that you can use to calculate the
perimeter of any rectangle. Use  for the length of
the rectangle, w for the width of the rectangle, and
The opposite
sides of a rectangle
are always the
same length.
P for the perimeter.
length and width. Explain how you can do this. Use the table in Question 5 to help you.
836 • Chapter 13 Area
© 2011 Carnegie Learning
9. You can determine the area of a rectangle without drawing it if you know the rectangle’s
10. Write a formula that you can use to calculate the area of any rectangle. Use  for
the length of the rectangle, w for the width of the rectangle, and A for the area of
the rectangle.
11. Can you determine the area of a rectangle if its perimeter is known?
Explain your reasoning.
12. Can you determine the perimeter of a rectangle if its area is known?
© 2011 Carnegie Learning
Explain your reasoning.
13.2 Area and Perimeter of Rectangles and Squares • 837
13. For each rectangle, either the length, width, or area is unknown. First, calculate the
value of the unknown measure. Then, calculate the perimeter.
a.
15 feet
21 feet
b.
Area:
48 square
millimeters
8 millimeters
c.
3.5 inches
© 2011 Carnegie Learning
Area: 15.75
square inches
838 • Chapter 13 Area
14. Calculate the perimeter and area of a rectangle that is 11 meters long and
5 meters wide.
a.Double the length and width of the rectangle. Calculate the perimeter of the
new rectangle.
b. What effect does doubling the length and width have on the perimeter?
c.Do you think that doubling the length and width will have the same effect on the
area? Explain your reasoning.
© 2011 Carnegie Learning
15. Calculate the area of the rectangle that had its dimensions doubled.
16. What effect does doubling the length and width have on the area?
13.2 Area and Perimeter of Rectangles and Squares • 839
Problem 2 A Square Rug
Tyson has also received several requests to create square-shaped rugs.
1. One square rug is seven feet long and seven feet wide. Draw a model of this rug on
the grid shown. Each square on the grid represents a square that is one foot long and
one foot wide.
3. What is the perimeter of this rug? Explain your calculation.
840 • Chapter 13 Area
© 2011 Carnegie Learning
2. What is the area of this rug? Explain your calculation.
4. Six different squares are drawn on the grid. The letters A through F name each square.
Each square on the grid represents a square that is one foot long and one foot wide.
C
A
B
E
F
D
5. Complete the table to show the length, width, area, and perimeter of each square.
Square
Length
(units)
Width
(units)
Perimeter
(units)
Area
(square units)
A
B
© 2011 Carnegie Learning
C
D
E
F
13.2 Area and Perimeter of Rectangles and Squares • 841
6. You can determine the perimeter of a square without drawing the square if you know
the length of one side of the square. Explain how you can do this. Use the table in
Question 5 to help you.
7. Write a formula that you can use to calculate the perimeter of any square.
Use s for the side length of the square and P for the perimeter.
8. You can determine the area of a square without drawing it if you know the length of
one side of the square. Explain how you can do this. Use the table in Question 5 to
help you.
9. Write a formula that you can use to calculate the area of any square. Use s for the
length of a side of the square and A for the area of the square.
10. Calculate the value of the unknown side length, area, and perimeter in each square shown.
a.
b.
Area: 169 square feet
842 • Chapter 13 Area
© 2011 Carnegie Learning
5 centimeters
11. Calculate the perimeter and area of a square that has a side length equal to 9 inches.
a.Double the side length of the square. Calculate the perimeter of the new square.
b.What effect does doubling the side length of a
square have on the perimeter?
Is your
reasoning about the
effect of changing side
lengths of a square, the
same or different than
your reasoning when you
considered side length
changes of a rectangle?
c.Do you think that doubling the side length of a
square will have the same effect on the area?
Explain your reasoning.
12. Calculate the area of a square that has a side length equal to
© 2011 Carnegie Learning
10 meters.
a.Double the length of the side of the square. Calculate the area of
the new square.
b.What effect does doubling the length of a side of a square have on the area?
13.2 Area and Perimeter of Rectangles and Squares • 843
Problem 3 A Brand New Floor
A carpeting company has been hired to install flooring on the first floor of a home.
A diagram of this first floor is shown.
35 feet
10 feet
10 feet
5 feet
Enclosed
porch
Dining
room
Kitchen
8 feet
a
Living
room
b
14 feet
10 feet
12 feet
1. Calculate the unknown lengths a and b.
Use what
you do know from
the diagram to
figure out what
you don't know.
2. The homeowners would like to install indoor/outdoor carpeting on the
enclosed porch. How many square feet of indoor/outdoor carpeting
© 2011 Carnegie Learning
will be needed?
844 • Chapter 13 Area
3. The homeowners would like to install wood flooring in the dining room. How many
square feet of wood flooring will be needed?
4. The homeowners would like to install tile in the kitchen. How many square feet of tile
will be needed?
5. The homeowners would like to install loop carpeting in the living room. How many
© 2011 Carnegie Learning
square feet of loop carpeting will be needed?
6. What is the total area of the first floor? Explain your reasoning.
13.2 Area and Perimeter of Rectangles and Squares • 845
7. An employee from the flooring company must now calculate the total cost of the
materials used to complete this job.
a.The carpeting for the porch costs $1.20 per square foot. Calculate the cost of
carpeting the enclosed porch.
b. The loop carpeting costs $0.84 per square foot. Calculate the cost of carpeting
the living room.
c. The wood flooring costs $4.50 per square foot. Calculate the cost of installing the
wood flooring in the dining room.
d. The tiling costs $4.25 per square foot. Calculate the cost of tiling the kitchen.
e. Calculate the total cost of the materials needed for the job.
g. After the homeowners saw the total cost of the flooring for the job, they decided that
the wood flooring was too expensive and decided to use the same loop carpeting in
the dining room as in the living room. Calculate the total cost of the flooring for the first
floor if the homeowners decide to buy the loop carpeting for the dining room.
Be prepared to share your solutions and methods.
846 • Chapter 13 Area
© 2011 Carnegie Learning
f. Do you think the total cost is accurate? Why or why not?
Boundary Lines
Area of Parallelograms
and Triangles
Learning Goals
Key Terms
In this lesson, you will:




 Calculate the area of parallelograms and triangles.
 Write a formula for the area of a parallelogram and
a triangle.
altitude of a parallelogram
height of a parallelogram
altitude of a triangle
height of a triangle
 Calculate the area of composite figures.
O
n you mark . . . get set . . . sail? Sailboat racing isn’t quite started like that,
but there is definitely a starting and finish line in this sport. Generally, each
sailboat has a crew that mans various sails. In fact, while you may think that
sailboats can only go in the direction of the wind, that notion is not quite true!
There are ways that sailboats can sail against the wind. In fact, there is almost an
entire science to sailing. How do you think that a sailboat can sail against the
© 2011 Carnegie Learning
wind? Do you think those same principles can be used for toy sailboats?
13.3 Area of Parallelograms and Triangles • 847
Problem 1 A Parallelogram Rug
Tyson has a special request from a client. The client would like a rug in the shape of a
non-rectangular parallelogram. A model of the rug is shown on the grid. Each square
on the grid represents a square that is one foot long and one foot wide.
1. Explain how you can create a rectangle from the figure shown so that the two figures
have the same area. Then, check your answer by demonstrating your method on a
2. What is the area of the rectangle from Question 1? Explain your calculation.
3. What is the area of the rug? Explain your calculation.
848 • Chapter 13 Area
© 2011 Carnegie Learning
separate sheet of grid paper. Draw your rectangle on top of the figure on the grid.
4. Tyson’s client requests another rug. A model of the new rug is shown on the grid.
Calculate the area of the rug. Explain your reasoning. Each square on the grid
represents a square that is one foot long and one foot wide.
Any side of a parallelogram is a base. Parallelogram EFGH shown is
drawn in different orientations. Each square on the grid represents a
square that is one foot long and one foot wide.
F
G
E
base
© 2011 Carnegie Learning
E
H
base
H
F
G
13.3 Area of Parallelograms and Triangles • 849
An altitude of a parallelogram is a line segment drawn from a vertex, perpendicular to
the line containing the opposite side. A height of a parallelogram is the perpendicular
distance from any point on one side to the line containing the opposite side.
F
G
altitude
base
E
E
H
So, the height
is just the length
of the altitude.
F
altitude
H
base
G
5. For each parallelogram, draw a segment that represents a height. Label the height
with its measure and label the base with its measure. Each square on the grid
© 2011 Carnegie Learning
represents a square that is one foot long and one foot wide.
6. Write a formula for the area of a parallelogram. Use b for the base of the
parallelogram, h for the height, and A for the area.
850 • Chapter 13 Area
7. For each parallelogram, the length of a base, the height, or the area is unknown.
Calculate the value of each unknown measure.
a.
9.5 feet
10 feet
b.
Area: 60 square meters
15 meters
Remember,
multiplication and
division are inverse
operations.
c.
Area: 28 square inches
© 2011 Carnegie Learning
3.5 inches
13.3 Area of Parallelograms and Triangles • 851
8. Tyson charges $20 per square foot of rug for a basic design. A client orders one basic
rectangular-shaped rug that is 6 feet long and 4 feet wide and one basic rug shaped
like a parallelogram that is not a rectangle with a base that is 8 feet long and a height
that is 3 feet. What is the total cost for the rugs? Explain your reasoning.
Problem 2 The Race Course
One of the typical shapes of a sailboat race course is triangular.
The course path is identified by buoys called marks. When the
course is triangular-shaped, the marks are located at the
vertices of the triangle.
A sample course with the marks numbered is shown. Each square on the grid represents a
square that is one tenth of a kilometer long and one tenth of a kilometer wide.
1
© 2011 Carnegie Learning
2
Wind
3
852 • Chapter 13 Area
1. How many grid squares in a row create an area that is one kilometer long and one
tenth of a kilometer wide?
2. How many grid squares are in an area that is one kilometer long and one kilometer wide?
3. Estimate the area enclosed by the course. Justify your estimate.
4. Is your area from Question 3 exact? Explain your reasoning.
© 2011 Carnegie Learning
5. Use two sides of the triangle to draw a parallelogram on the grid.
6. Calculate the area of the parallelogram you drew.
7. Can you calculate the exact area of the triangle by using the area of the
parallelogram? Why or why not?
13.3 Area of Parallelograms and Triangles • 853
8. Calculate the exact area enclosed by the triangular course.
9. How does the exact area enclosed by the triangular-shaped course compare to
the estimate?
But why
are the two
triangles that form
a parallelogram
congruent?
10. How does the area of the parallelogram relate to the
area of the triangle?
11. Consider the race course shown on the grid. Each square on the
grid represents a square that is one tenth of a kilometer long and
one tenth of a kilometer wide. Calculate the area enclosed by
the course.
1
2
© 2011 Carnegie Learning
3
854 • Chapter 13 Area
12. What information about the triangle did you need to calculate the area in Question 11?
Any side of a triangle is a base.
Triangle KYM is the same triangle drawn in three different orientations.
M
K
Y
Y
base
M
M
base
K
K
base
Y
An altitude of a triangle is a line segment drawn from a vertex perpendicular to a line
containing the opposite side. A height of a triangle is the perpendicular distance from a
vertex to the line containing the base.
M
© 2011 Carnegie Learning
K
Y
altitude
altitude
altitude
Y
base
M
M
base
K
K
base
Y
13. Write a formula that you can use to calculate the area of any triangle. Use b for the
length of the base, h for the height, and A for the area of the triangle.
13.3 Area of Parallelograms and Triangles • 855
14. Determine the base and height of triangles KYM, MYK, KMY. Then calculate the area
of each triangle.
M
K
Y
Y
base
M
M
base
K
K
base
Y
15. Describe what happens to the height of a triangle as the length of the base changes
when the area remains the same.
16. For each triangle, the length of the base, the height, or the area is unknown. Calculate
the value of each unknown measure.
a.
Area: 60 square meters
b.
6 yards
8 yards
856 • Chapter 13 Area
© 2011 Carnegie Learning
24 meters
c.
Area: 42 square inches
6 inches
17. The original race course is shown, but now the lengths of the legs of the race are
given. If a boat must complete the course once, how long is the race?
2
1.6 kilometers
1
1.6 kilometers
1.6 kilometers
Wind
3
© 2011 Carnegie Learning
18. What geometric name is given to this measurement?
19. In sailboat races, it is common for a boat to have to go around a course more than
once or revisit a leg of the course more than once. Suppose that to complete the
race, a boat must sail to the marks in the following order: 1, 2, 3, 1, 3, 1, 2, 3, 1, 3.
How long is this race?
13.3 Area of Parallelograms and Triangles • 857
20. If a boat is competing in a race, do you think that the boat will travel more than or less
than the race length you calculated in Question 19.
Problem 3 Boundary Lines
1. Each square on the grid formed by connecting four closely positioned dots represents
a square that is 1 unit long and 1 unit wide. Line KN is drawn parallel to line PR.
K
M
P
N
R
a. Calculate the area of triangle KPR.
c. Calculate the area of triangle NPR.
d. Compare the areas of triangles KPR, MPR, and NPR.
e. Compare the bases of triangles KPR, MPR, and NPR.
858 • Chapter 13 Area
© 2011 Carnegie Learning
b. Calculate the area of triangle MPR.
f. Compare the heights of triangles KPR, MPR, and NPR.
g.What conclusion can be made about triangles that share the same base, or have
bases of equal measure, and also have equal heights?
2. Use the conclusion you made in Question 1, part (g) to solve this problem.
A sister and brother inherit equal amounts of property; however, the boundary line
separating their land is not straight. Your job is to draw a new boundary line that is
straight and keeps the property division equal. Explain how you solved the problem.
A
Sister’s
Land
© 2011 Carnegie Learning
B
Upper boundary line
Brother’s
Land
Lower boundary line
Also think
about the area
relationships you already
established about
parallelograms and
triangles.
13.3 Area of Parallelograms and Triangles • 859
Problem 4 Composite Figures
1. The figure shown is composed of a rectangle and triangles.
A
R
T
20 ft
E
60 ft
C
a. Describe a strategy that can be used to compute the area of the shaded region.
It really
helps to think
about a few
strategies before
jumping right in!
b Calculate the area of rectangle RECT.
© 2011 Carnegie Learning
c. Calculate the area of triangle AEC.
860 • Chapter 13 Area
d. Calculate the area of the shaded region.
2. The figure shown is composed of rectangles and triangles.
R
H
E
X
A
1.5"
N
O
T
1"
G
C
5"
1"
a. Describe two different strategies that can be used to compute the area of the
© 2011 Carnegie Learning
entire shaded region.
13.3 Area of Parallelograms and Triangles • 861
b. Use one of your strategies to calculate the area of the shaded region.
© 2011 Carnegie Learning
Be prepared to share your solutions and methods.
862 • Chapter 13 Area
The Keystone
Effect
Area of Trapezoids
Learning Goals
Key Terms
In this lesson, you will:




 Calculate the area of trapezoids.
 Write a formula for the area of a trapezoid.
 Calculate the area of composite figures.
bases of a trapezoid
legs of a trapezoid
altitude of a trapezoid
height of a trapezoid
I
n most classrooms, a projection screen is hung above the blackboard. Generally,
the screen is located higher than where the projector sits. To view images on the
screen, the projector must be tilted upward. This tilting can cause “keystoning,”
which is a distortion of the image. A normal image and possible distorted image are shown.
Four things to remember when using the Internet:
© 2011 Carnegie Learning
• Never share your personal information, such as
addresses, phone numbers, or photographs, with
online friends.
• Never agree to meet someone face-to-face you
met online.
• Never respond to messages
or bulletin boards that make
you feel uncomfortable.
• People you meet online may
not be who they say they are.
13.4 Area of Trapezoids • 863
Problem 1 How Distorted?
1. Describe how the normal image from the lesson opener has been distorted.
2. Describe the shapes formed by the normal image and the distorted image.
3. Which image do you think has a larger area? Explain your reasoning.
4. The normal image and the distorted image are shown on the grid. Each square on the
grid represents a square that is four inches long and four inches wide. Calculate the
© 2011 Carnegie Learning
area of each image and write it in the center of the image.
5. How do the areas of the images compare?
864 • Chapter 13 Area
6. Is your area of the distorted image exact? Explain your reasoning.
7. Consider the distorted image in Question 4. How can you use the area formulas you
already know to calculate the exact area of this image?
8. Calculate the exact area of the distorted image.
© 2011 Carnegie Learning
Pay attention
to the scale on
the grid!
13.4 Area of Trapezoids • 865
9. How do the exact areas of the normal image and the distorted image compare?
10. Consider the distorted image again. Suppose that you make an exact copy of this
image, flip it vertically, and move it next to the image as shown.
a.What is the geometric figure that is formed from these images?
b. Use a formula to calculate the area of the parallelogram. Then, use that area to
calculate the area of the distorted image.
© 2011 Carnegie Learning
You did
something just
like this when you
related the areas of
parallelograms and
triangles.
866 • Chapter 13 Area
11. Was it easier to calculate the area of the distorted image by using your method in
Question 7 or by using the method in Question 10? Explain your reasoning.
The distorted image is a trapezoid. The parallel sides of the trapezoid are called the bases
of the trapezoid. Non-parallel sides are the legs of the trapezoid.
Trapezoid TRAP is the same trapezoid drawn in different orientations.
base
P
T
base
T
A
R
R
base
P
P
A
A
R
T
© 2011 Carnegie Learning
base
base
base
base
base
T
R
A
P
13.4 Area of Trapezoids • 867
An altitude of a trapezoid is a line segment drawn from a vertex perpendicular to a line
containing the opposite side. A height of a trapezoid is the perpendicular distance from a
vertex to the line containing the base.
base
P
T
A
altitude
T
base
R
altitude
R
P
A
P
A
R
T
altitude
base
base
altitude
T
R
A
P
12. Consider the trapezoid shown. Suppose that you make
an exact copy of this trapezoid, flip it vertically, and
move it next to the trapezoid as shown. Label the bases
of the trapezoid on the right.
Can you
prove to me that
this figure is a
parallelogram?
b1
b2
a. Write a formula for the area of the entire figure.
b.Write a formula for the area of one of the trapezoids.
Explain your reasoning.
868 • Chapter 13 Area
© 2011 Carnegie Learning
h
13. For each trapezoid, either a height, the length of one base, or the area is unknown.
Determine the value of each unknown measure.
a.
22 millimeters
8 millimeters
6 millimeters
b.
3 feet
Area: 25 square feet
7 feet
c.
© 2011 Carnegie Learning
6 meters
Area: 45 square meters
9 meters
13.4 Area of Trapezoids • 869
14. The projector in Problem 1 was tilted differently to create the distorted image
shown. Each square on the grid represents a square that is four inches long and
four inches wide.
Normal
Image
Distorted
Image
a. What is the area of the distorted image?
b.How does the area of the distorted image compare to the area of the
normal image?
Problem 2 Composite Figures
80 cm
R
22 cm
T
30 cm
E
870 • Chapter 13 Area
C
© 2011 Carnegie Learning
1. The figure shown is composed of a rectangle and four congruent trapezoids.
a. Describe a strategy that can be used to compute the area of the shaded region.
b. Calculate the area of rectangle RECT.
© 2011 Carnegie Learning
c. Calculate the area of a trapezoid.
d. Calculate the area of the shaded region.
13.4 Area of Trapezoids • 871
2. The figure is composed of a rectangle and a regular hexagon.
The length of each side of the hexagon is 2 centimeters.
4
2 cm
3.5
a.Describe two strategies that can be used to compute the area of the shaded
region.
© 2011 Carnegie Learning
b. Calculate the area of the shaded region.
872 • Chapter 13 Area
Talk the Talk
Write the area formula for each figure.
1.
2.
w
s
<
s
Area of a Rectangle Formula:
3.
Area of a Square Formula:
4.
h
h
b
Area of a Parallelogram Formula:
5.
b
Area of a Triangle Formula:
b1
h
b2
© 2011 Carnegie Learning
Area of a Trapezoid Formula:
Be prepared to share your solutions and methods.
13.4 Area of Trapezoids • 873
© 2011 Carnegie Learning
874 • Chapter 13 Area
Go Fly a Kite
Area of Rhombi and Kites
Learning Goals
In this lesson, you will:
 Calculate the area of rhombi and kites.
 Use formulas to compute the area of rhombi and kites.
 Calculate the area of composite figures.
W
hat hobby’s history is older than our current calendar? If you said kites,
you’d be correct. It is thought that kites were created around 2800 years ago in
China. Early kites were made of silk and had tails. From China, the kite traveled to
India where it evolved to a different type of kite called a fighter kite, or patang.
Fighter kiting is still in existence today. Competition can be tough as each kite
flyer tries to ground his or her opponent’s kite through various maneuvers. And
let’s not forget one of the most famous kites that may or may not have been
flown. In 1750, Benjamin Franklin proposed flying a kite during a lightning storm to
prove that electricity existed in lightning. Have you ever flown a kite? How do you
think Benjamin Franklin could have used a kite to prove electricity exists in
© 2011 Carnegie Learning
lightning?
13.5 Area of Rhombi and Kites • 875
Problem 1 Area of a Rhombus
Recall that a rhombus is a quadrilateral with all sides congruent.
Mr. Gram asked his math students to sketch a parallelogram. These are two of his
students’ sketches.
Molly’s Sketch
James’s Sketch
3 cm
E
F
4 cm
A
B
2 cm
D
3 cm
3 cm
2 cm
4 cm
C
H
3 cm
G
1. How are the two sketches similar?
2. How are the two sketches different?
3. Mr. Gram told his class that both sketches are examples of parallelograms, but
James’s sketch is an example of a special parallelogram. Why would James’s sketch
© 2011 Carnegie Learning
be considered a special parallelogram?
876 • Chapter 13 Area
4. Mr. Gram wrote two statements on the board.
• All parallelograms are rhombi.
• All rhombi are parallelograms.
Are both of these statements true? Explain your reasoning.
5. Sketch a parallelogram that is not a rhombus.
© 2011 Carnegie Learning
6. Sketch a rhombus that is not a parallelogram.
7. Molly concluded that all squares must be rhombi and parallelograms. Is she correct?
Explain your reasoning.
13.5 Area of Rhombi and Kites • 877
8. Mr. Gram told Molly that he is thinking of a quadrilateral that is either a square or a
rhombus that is not a square. He wants Molly to guess which quadrilateral he is
thinking of, and he allows her to ask one question about the quadrilateral. Which
question should she ask?
9. Since all rhombi are also parallelograms, what formula can be used to calculate
the area of a rhombus? Use b for the length of the base, h for the height, and A for
the area.
Problem 2 Area of a Kite
Recall that a kite is a quadrilateral with two pairs of consecutive congruent sides with
opposite sides that are not congruent.
Kite ABCD
B
A
C
1.5 cm
4 cm
D
878 • Chapter 13 Area
© 2011 Carnegie Learning
4 cm
1.5 cm
1. Mr. Gram sketched the kite shown. He asked his
So, it can't
just look like a
kite. In mathematics,
kite has a very specific
definition.”
students to add a line segment such that it would
divide the kite into two familiar figures.
Molly’s kite
B
4 cm
1.5 cm
A
C
1.5 cm
4 cm
D
James’s kite
B
4 cm
1.5 cm
A
C
1.5 cm
4 cm
D
If Mr. Gram asked you to determine the area of the kite, would you rather use Molly’s
kite or James’s kite? Explain.
© 2011 Carnegie Learning
2. Describe a strategy that can be used to compute the area of the kite.
3. If you are using Molly’s kite to calculate the area of the kite, what additional
information would you need?
13.5 Area of Rhombi and Kites • 879
4. If you are using James’s kite to calculate the area of the kite, what additional
information would you need?
5. How would the area of Molly’s kite compare to the area of James’s kite?
6. Given:
B
AC = 5 cm
BD = 2.5 cm
AE = 1.1 cm
BE = 1.25 cm
CE = 3.9 cm
DE = 1.25 cm 4 cm
1.5 cm
A
E
1.5 cm
C
4 cm
D
Calculate the area of kite ABCD using both Molly’s and James’s strategies.
Did your
calculations verify
your predictions about
the area of Molly's kite
compared to the area
of James's kite?
880 • Chapter 13 Area
© 2011 Carnegie Learning
Problem 3 Composite Figures
1. The figure shown is composed of a rectangle and a kite.
B
A
E
C
D
Describe a strategy that can be used to compute the area of the shaded region.
2. How do you think the areas of the triangles in the kite compare to the areas of the
© 2011 Carnegie Learning
triangles in the shaded region?
3. How do you think the area of the kite compares to the area of the shaded region?
13.5 Area of Rhombi and Kites • 881
4. Given:
AC = 10 m
BE = 4 m
Calculate the area of the kite.
5. Calculate the area of the rectangle.
© 2011 Carnegie Learning
6. Calculate the area of the shaded region.
Be prepared to share your solutions and methods.
882 • Chapter 13 Area
Street Signs
Area of Regular Polygons
Learning Goals
Key Terms
In this lesson you will:
 congruent polygons
 apothem
 Calculate the area of regular polygons.
 Write a formula for the area of a regular polygon.
 Calculate the area of composite figures.
H
ave you ever noticed that every stop sign looks exactly the same, every yield
sign looks exactly the same, and so on? This is because the Federal Highway
Administration has standards that indicate the exact sizes and colors of roadway
© 2011 Carnegie Learning
signs. Most of the sign shapes are polygons.
13.6 Area of Regular Polygons • 883
Problem 1 How Big is that Sign?
The specifications for the smallest possible yield sign are shown.
30 in.
60
60o
o
YIELD
30 in.
30 in.
60o
1. What is special about the triangle that forms the yield sign?
The specifications for a “Do Not Enter” sign are shown.
30 in.
90o
90o
DO NOT
30 in.
30 in.
ENTER
90o
90o
30 in.
The specifications for a stop sign are shown.
STOP
12.4 in.
12.4 in.
135o
135o
135o
135o
12.4 in.
12.4 in.
135o
135o
12.4 in.
135o
135o
12.4 in.
884 • Chapter 13 Area
12.4 in.
12.4 in.
© 2011 Carnegie Learning
2. What is special about the quadrilateral that forms the “Do Not Enter” sign?
3. What is special about the octagon that forms the stop sign?
The polygons in Questions 1 through 3 are special polygons called regular polygons. Two
other possible sizes for a yield sign are shown.
48 in.
36 in.
60
60
o
60o
60o
o
48 in.
36 in.
48 in.
36 in.
60o
60o
4. Are these signs regular polygons? What can you conclude about all regular triangles?
5. The yield sign from Question 1 is shown with its approximate height. Calculate the
approximate area of the yield sign.
30 in.
YIELD
© 2011 Carnegie Learning
26 in.
6. Calculate the approximate area of the “Do Not Enter” sign from Question 2.
13.6 Area of Regular Polygons • 885
When two polygons are exactly the same size and exactly the same shape, the polygons
are said to be congruent polygons.
7. To calculate the area of the stop sign from Question 3, you can use the fact that a
regular polygon can be divided into triangles that are all exactly the same size and
same shape. The bases of the triangles are the sides of the polygon as shown. In this
case, the height of each triangle is approximately 15 inches. Calculate the area of the
stop sign. Round your answer to the nearest tenth if necessary. Explain your reasoning.
12.4 in.
15 in.
The height of the triangle in the stop sign in Question 7 is
polygon is the perpendicular distance from the center of the
regular polygon to a side of the regular polygon.
8. Draw a segment that represents an apothem on
each regular polygon shown. The center of the
polygon is marked by a point.
886 • Chapter 13 Area
Many dictionaries
say the correct
pronunciation
of apothem is
AP-uh-thum.
© 2011 Carnegie Learning
the apothem of the octagon. The apothem of a regular
9. The hexagon shown is a regular hexagon. Calculate the area of the hexagon. Explain
your reasoning.
40 cm
34.6 cm
10. The heptagon shown is a regular heptagon. Calculate the area of the heptagon.
Explain your reasoning.
8m
© 2011 Carnegie Learning
8.3 m
13.6 Area of Regular Polygons • 887
11. Explain how you can calculate the area of a regular polygon if you know the length of
the apothem and the length of each side.
12. Write a formula for the area of a regular polygon with n sides. Use a for the length of
the apothem and  for the length of one side of the polygon.
Problem 2 Perimeter and Apothems
(  )
1. Lily claims the formula for determining the area of a regular polygon is A 5 ​__
​ 1 ​  a  ​n,
2
where  is the length of a side, a is the apothem, and n is the number of sides.
​ 1 ​ Pa,
Molly claims the formula for determining the area of a regular polygon is A 5 __
2
where P is the perimeter of the polygon and a is the apothem.
Who is correct? Explain your reasoning.
© 2011 Carnegie Learning
888 • Chapter 13 Area
2. Emma thinks the definition for a regular polygon is too long and it should be
shortened. She believes that if a polygon has all sides equal in length, then all angles
will always be equal in measure.
a. What are two examples Emma could use to justify her conclusion?
b. Is Emma correct? Justify your conclusion.
3. Jath also thinks the definition for a regular polygon is too long. He states that if a
polygon has all angles of equal measure, then all sides will always be equal in length.
© 2011 Carnegie Learning
a. What are two examples Jath could use to justify his conclusion?
b. Is Jath correct? Justify your conclusion.
13.6 Area of Regular Polygons • 889
Problem 3 Calculate the Area
1. The length of one side of a regular nonagon is 24 feet, and the length of the apothem
is approximately 33 feet. Calculate the area of the regular nonagon.
2. The side length of the largest possible stop sign is 20 inches, and the length of the
apothem is approximately 24.1 inches.
a. What is the area of the largest possible stop sign?
b. The side length of the smallest possible stop sign is 9.9 inches, and the length of
the apothem is approximately 12 inches. What is the area of the smallest possible
stop sign?
of the smallest possible stop sign? Explain your reasoning.
890 • Chapter 13 Area
© 2011 Carnegie Learning
c. How many times larger is the area of the largest possible stop sign than the area
Problem 4 How Big Is the Pentagon?
The United States Defense Department is located in a building called the Pentagon in
Arlington, Virginia. This regular pentagonal-shaped building has 17.5 miles of corridors.
Each side of the building is approximately 921 feet (307 yards) long, and the apothem of
the pentagon is approximately 633.8 feet (211.27 yards).
1. Determine the approximate area of the ground level of the Pentagon in square yards.
A football
field is 50 yards
wide and 100
yards long.
2. Determine the area of a football field.
© 2011 Carnegie Learning
3. How does the area of a football field compare to the area on
the ground level of the Pentagon?
13.6 Area of Regular Polygons • 891
Problem 5 Composite Figures
The figure shown is composed of two regular hexagons.
20 mm
12 mm
10 mm
6 mm
1. Describe a strategy that can be used to compute the area of the shaded region.
Be prepared to share your solutions and methods.
892 • Chapter 13 Area
© 2011 Carnegie Learning
2. Calculate the area of the shaded region.
Think back to all the
composite figures you
have worked with in this
chapter. Did you use
similar strategies to
calculate the area of
the shaded region
in each?
Chapter 13 Summary
Key Terms
protractor (13.1)
compass (13.1)
straightedge (13.1)
sketch (13.1)
draw (13.1)
construct (13.1)
triangle (13.1)
equilateral triangle (13.1)
isosceles triangle (13.1)
scalene triangle (13.1)
equiangular triangle (13.1)
acute triangle (13.1)
right triangle (13.1)
obtuse triangle (13.1)
quadrilateral (13.1)
opposite sides (13.1)
consecutive sides (13.1)
square (13.1)
rectangle (13.1)
rhombus (13.1)
parallelogram (13.1)
kite (13.1)
trapezoid (13.1)
isosceles trapezoid (13.1)
polygon (13.1)
regular polygon (13.1)
irregular polygon (13.1)
pentagon (13.1)
hexagon (13.1)
heptagon (13.1)
octagon (13.1)
nonagon (13.1)
decagon (13.1)
altitude of a parallelogram
(13.3)
height of a parallelogram
(13.3)
altitude of a triangle (13.3)
height of a triangle (13.3)
bases of a trapezoid (13.4)
legs of a trapezoid (13.4)
altitude of a trapezoid (13.4)
height of a trapezoid (13.4)
congruent polygons (13.6)
apothem (13.6)
Sketching and Drawing Polygons
When you sketch a geometric figure, the figure is created
without the use of tools. When you draw a geometric
figure, the figure is created with the use of tools such
as a ruler, a straightedge, a compass, or a protractor.
© 2011 Carnegie Learning
A protractor can be used to approximate the
measure of an angle. A compass is a tool used to
create arcs and circles. A straightedge is a ruler with
no numbers.
Think the size
or shape of your brain
matters? It doesn't. In fact
your brain keeps growing and
changing throughout your life
every time you learn
something new!
Example
A sketch and a drawing of a parallelogram are shown.
sketch
drawing
Chapter 13 Summary • 893
Naming Polygons and their Sides and Angles
A polygon is a closed figure that is formed by joining three or more line segments at their
endpoints. A regular polygon is a polygon with all sides congruent and all angles
congruent. An irregular polygon is a polygon that is not regular. Polygons are named using
capital letters representing the vertices, listed in clockwise or counterclockwise order.
Example
Right triangle XYZ is shown.
X
Z
Y
Triangle XYZ can also be named nYZX, nZXY, nXZY, nZYX, or nYXZ. The sides of nXYZ
are XY, YZ, and ZX. The angles of nXYZ are ∠X, ∠Y, and ∠Z.
Sorting Polygons into Categories
●
Triangles have 3 sides.
●
Quadrilaterals have 4 sides.
●
Pentagons have 5 sides.
●
Hexagons have 6 sides.
●
Heptagons have 7 sides.
●
Octagons have 8 sides.
●
Nonagons have 9 sides.
●
Decagons have 10 sides.
894 • Chapter 13 Area
© 2011 Carnegie Learning
Polygons are classified by the number of sides they have.
Some quadrilaterals have special names because of their specific qualities. A square is a
quadrilateral with all sides congruent and all angles congruent. A rectangle is a
quadrilateral with opposite sides congruent and all angles congruent. A rhombus is a
quadrilateral with all sides congruent. A parallelogram is a quadrilateral with both pairs of
opposite sides parallel. A kite is a quadrilateral with two pairs of consecutive congruent
sides with opposite sides that are not congruent. A trapezoid is a quadrilateral with exactly
one pair of parallel sides. An isosceles trapezoid is a trapezoid whose non-parallel sides
are congruent.
Example
These regular polygons have been sorted into two groups. One group contains regular
polygons whose opposite sides are parallel and the other group contains regular polygons
without any opposite sides that are parallel.
© 2011 Carnegie Learning
Regular polygons with opposite sides parallel. Regular polygons without opposite sides parallel.
Square
Regular Hexagon
Regular Triangle
Regular Pentagon
Regular Octagon
Regular Decagon
Regular Heptagon
Regular Nonagon
Notice that all of the regular polygons with opposite sides parallel have an even number of
sides. Similarly, all of the regular polygons without opposite sides parallel have an odd
number of sides.
Chapter 13 Summary • 895
Calculating the Perimeter and Area of a Rectangle
To calculate the perimeter, P, of a rectangle, use the formula P 5 2,1 2w, where ,
represents the length of the rectangle and w represents the width of the rectangle. To
calculate the area, A, of a rectangle, use the formula A 5 ,w.
Example
5 ft
16 ft
P 5 2, 1 2w 5 2(16) 1 2(5) 5 32 1 10 5 42 feet
A 5 ,w 5 16(5) 5 80 square feet
The perimeter of the rectangle is 42 feet, and the area of the rectangle is 80 square feet.
Calculating the Perimeter and Area of a Square
To calculate the perimeter, P, of a square, use the formula P 5 4s, where s is the side
length of the square. To calculate the area, A, of a square, use the formula
A 5 s 3 s or A 5 s2.
Example
12 m
P 5 4s 5 4(12) 5 48 meters
A 5 s · s 5 s2 5 122 5 144 square meters
The perimeter of the square is 48 meters, and the area of the square is 144 square meters.
896 • Chapter 13 Area
© 2011 Carnegie Learning
12 m
Calculating the Area of Composite Figures
Composite figures are figures that are composed of multiple geometric shapes. To
determine the area of a composite figure, it is usually best to separate the figure into its
composite shapes, calculate the area of each shape, and then add these areas together to
determine the total area.
Example
The figure shown is a composite figure consisting of one square and two congruent rectangles.
10 yards
2 yards
6 yards
2 yards
6 yards
The length of each side of the square is 6 yards. The area of the square is A 5 s · s 5 s2 5
62 = 36 square yards. Each rectangle has a length of 4 yards and a width of 2 yards. The
area of one rectangle is A 5 ,w 5 4(2) 5 8 square yards. The total area of the composite
figure is 36 1 2(8) 5 36 1 16 5 52 square yards.
Calculating the Area of a Parallelogram
To calculate the area, A, of a parallelogram, use the formula A 5 bh, where b represents the
length of the base of the parallelogram and h represents the height of the parallelogram.
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Example
10.5 m
22 m
A 5 bh
5 22(10.5)
5 231 square meters
The area of the parallelogram is 231 square meters.
Chapter 13 Summary • 897
Calculating the Area of a Triangle
To calculate the area, A, of a triangle, use the formula A 5 __
​ 1 ​ bh, where b represents the
2
length of the base of the triangle and h represents the height of the triangle.
Example
7 cm
11 cm
A 5 __
​ 1 ​ bh
2
1
__
5 ​    ​(11)(7)
2
5 38.5 square centimeters
The area of the triangle is 38.5 square centimeters.
Calculating the Area of a Trapezoid
To calculate the area, A, of a trapezoid, use the formula A 5 __
​ 1 ​ (b1 1 b2)h, where b1 and b2
2
represent the lengths of the two bases and h represents the height of the trapezoid.
Example
30 yd
65 yd
A 5 __
​ 1 ​ (b1 1 b2)h
2
1
5 __
​   ​ (35 1 65)(30)
2
5 1500 square yards
The area of the trapezoid is 1500 square yards.
898 • Chapter 13 Area
© 2011 Carnegie Learning
35 yd
Calculating the Area of a Rhombus
All rhombi are parallelograms. Therefore, to calculate the area of a rhombus, use the area
formula for a parallelogram.
Example
Each side of the rhombus shown has a length of 10 inches.
8.5 in
10 in
A 5 bh
5 10(8.5)
5 85 square inches
© 2011 Carnegie Learning
The area of the rhombus is 85 square inches.
Chapter 13 Summary • 899
Calculating the Area of a Kite
A kite can be thought of as a composite figure consisting of two triangles. To calculate the
area of a kite, calculate the area of each triangular region and then add the two areas
together.
Example
7 ft
12 ft
The kite is composed of two triangles, each with a base length of 12 feet and a height of
3.5 feet.
The area of one triangle is:
A 5 __
​ 1 ​ bh
2
1
__
5 ​    ​(12)(3.5)
2
5 21 square feet
© 2011 Carnegie Learning
The area of the kite is 2(21) or 42 square feet.
900 • Chapter 13 Area
Calculating the Area of a Regular Polygon
The apothem of a regular polygon is the perpendicular distance from the center of the
polygon to a side of the polygon. To calculate the area, A, of a regular polygon, use the
formula A 5 __
​ 1 ​ Pa, where P represents the perimeter of the polygon and a represents the
2
length of the apothem.
Example
13.8 m
20 m
The perimeter of the pentagon is 5(20) or 100 meters.
A 5 __
​ 1 ​ Pa
2
1
__
5 ​    ​(100)(13.8)
2
5 690 square meters
© 2011 Carnegie Learning
The area of the pentagon is 690 square meters.
Chapter 13 Summary • 901
© 2011 Carnegie Learning
902 • Chapter 13 Area