Area of Polygons and Circles
Play Area
ACTIVITY
5.3
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
Pictured below is an aerial view of a playground. An aerial view is
the view from above something. Decide what piece of playground
equipment each figure below represents.
A
B
C
D
E
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G
F
1. Look at the shape of each figure, and write the name of the
playground equipment next to each letter.
A.
B.
C.
D.
E.
F.
G.
Unit 5 • Geometry
253
ACTIVITY 5.3
Area of Polygons and Circles
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Quickwrite, Self
Revision/Peer Revision, Group Presentation
My Notes
To plan the layout of a playground, a designer must know how
much area each piece of playground equipment takes up.
2. The aerial view of the playground contains many polygons.
a. What is a polygon?
b. Is a circle a polygon? Explain your reasoning.
3. Complete the table by listing all the geometric shapes you can
identify in each figure in the aerial view of the playground.
Figure
Geometric Shape(s)
A
B
C
D
F
G
4. Explain how you would find the area of Figure E.
5. Now consider the parallelogram in the aerial view of the
playground that is not also a rectangle. List some characteristics
of a parallelogram.
Page 255 contains shapes you will work with in this activity. Cut
each one out as you start the question that uses that shape.
254
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© 2010 College Board. All rights reserved.
E
Area of Polygons and Circles
Play Area
ACTIVITY 5.3
continued
Two Congruent Parallelograms (Cut these out when you start Question 6.)
Two Congruent Triangles (Cut these out when you start Question 10.)
© 2010 College Board. All rights reserved.
Two Congruent Trapezoids (Cut these out when you start Question 13.)
Circle (Cut this out when you start Question 19.)
Unit 5 • Geometry
255
This page is blank.
Area of Polygons and Circles
ACTIVITY 5.3
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Look for a Pattern, Self Revision/Peer Revision, Debriefing
My Notes
6. Cut out one of the two congruent parallelograms on page 255.
Then cut that parallelogram once in such a way that the two
pieces can be put together to form a rectangle.
a. Use a ruler to measure the rectangle and find its area.
ACADEMIC VOCABULARY
Two or more figures that
have the same shape and
size are congruent.
b. Sketch the rectangle and record your measurements in the
My Notes space.
7. Explain how the lengths of the base and height of the rectangle
you formed relate to those of the original parallelogram. (Use
the second parallelogram to compare.)
© 2010 College Board. All rights reserved.
8. Find a relationship between the base, height, and area of a
parallelogram. Describe that relationship using words, symbols,
or both.
The height of a figure is always
drawn perpendicular to its base.
Perpendicular lines (⊥) meet to
form right angles.
9. The hexagon in the aerial view of the playground is made up of
triangles and pentagons. List some characteristics of each figure.
a. hexagon
b. triangle
c. pentagon
Unit 5 • Geometry
257
ACTIVITY 5.3
Area of Polygons and Circles
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Create Representations, Quickwrite
My Notes
ACADEMIC VOCABULARY
The altitude of a triangle
is a perpendicular line
segment from a vertex to
the line containing the
opposite side. The measure
of an altitude is height.
10. Find the congruent triangles on page 255.
• Cut out one of the two triangles.
• Label one of its sides b.
• Draw the altitude of the triangle by drawing a line segment
perpendicular to side b. Label the segment h.
• Cut out the second triangle.
• Place the two triangles together to form a parallelogram
whose base is the side you labeled b.
11. How does the area of each triangle compare to the area of the
parallelogram from Question 10? Explain below.
12. Using words, symbols, or both, describe a method for finding
the area of a triangle.
MATH TERMS
The parallel sides of a trapezoid
are called the bases.
The two sides that are not
parallel are called the legs.
READING MATH
Sometimes subscripts are used
to label segments.
Another shape seen in the aerial view
of the playground looks like the figure
at right. This figure is called a trapezoid.
13. Find the congruent trapezoids on page 255.
• Cut out the two congruent trapezoids.
b1
• On the inside of each figure label the
bases as b1 and b2 as shown at right.
b2
• Draw in the height of the trapezoid
and label it h.
• Form a parallelogram by turning one of the trapezoids so
that its short base lines up with the long base of the other
trapezoid. The long legs of the trapezoids will be adjacent.
b1 is read as “b sub 1” and
represents one base of the
trapezoid.
b2 is read as “b sub 2” and
represents a second base of the
trapezoid.
258
SpringBoard® Mathematics with MeaningTM Level 1
© 2010 College Board. All rights reserved.
A trapezoid is a quadrilateral
with exactly one pair of parallel
sides.
Area of Polygons and Circles
ACTIVITY 5.3
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Group Presentation, Self Revision/Peer Revision, Quickwrite
My Notes
14. What is the height of the parallelogram? How does it compare
to the height of the original trapezoid?
15. What is the length of the base of the parallelogram? How does
it compare to the base of the trapezoid?
16. What is the area of the parallelogram?
17. What is the area of one of the trapezoids used to form the
parallelogram?
© 2010 College Board. All rights reserved.
18. A pentagon is another polygon in the aerial view of the playground. Use what you have learned about finding the area of
rectangles, triangles, parallelograms, and trapezoids to describe
how to find the area of this pentagon.
The last shape found in the aerial view of the playground on the
first page of this activity is a circle. Use the circle on page 255 to
complete the following questions.
19. Cut the circle into eight congruent pie-shaped
pieces. Arrange your eight pieces using the
alternating pattern shown at right.
Unit 5 • Geometry
259
ACTIVITY 5.3
Area of Polygons and Circles
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Guess and Check, Group Presentation, Debriefing
My Notes
20. Sketch the shape you just made with the circle pieces. What
shape does it resemble?
21. In your sketch, draw and label the height of the figure. What
part of the circle does the height represent?
22. What other part of the circle is about the length needed to
find the area of the shape you named in Question 21?
Right angles are often identified
with a small square in the corner
of the angle.
24. The dimensions of some of the pieces of playground equipment
are shown with their drawings below. Find the area of each figure.
Explain how you found each area.
a. Figure E
2 feet
10 feet
b. Figure G
2 feet
4 feet
260
SpringBoard® Mathematics with MeaningTM Level 1
© 2010 College Board. All rights reserved.
23. Using words, symbols, or both, describe how you can now
find the area of the circle.
Area of Polygons and Circles
ACTIVITY 5.3
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Group Presentation
My Notes
c. Figure A
1 ft
0.5 ft
1 ft
1 ft
2 ft
2 ft
8 ft
d. Figure F
3.46 ft
1 ft
2 ft
2 ft
2 ft
3 ft
2 ft
© 2010 College Board. All rights reserved.
3.46 ft
This aerial view is composed of three triangles and three
pentagons. Each of the outside segments measures 3.46 feet
while each of the inside segments measures 2 feet.
Recall that we can approximate π
22 .
as either 3.14 or ___
7
e. Figure B
6 ft
Unit 5 • Geometry
261
ACTIVITY 5.3
Area of Polygons and Circles
continued
Play Area
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
25. Based on the dimensions given for the other figures and the
location of Figures C and D on the playground, make an
estimate of the area of Figures C and D. Explain how you
arrived at your estimate.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show
your work.
22 to find the area of the circle.
5. Use π ≈ ___
7
Find the area of each figure. Remember to label
your answer.
1.
3 in.
14 in.
6. Find the area.
7 in.
14.4 in.
2.
7.8 in.
15 cm
3. Draw the figure, and then find the area of
a triangle with a base that measures 8.3 cm
and a height that measures 7.2 cm.
4. Find the area.
12 cm
8 cm
15.4 cm
262
SpringBoard® Mathematics with MeaningTM Level 1
7. Mikel is going to build a doghouse for
his new puppy. The floor’s shape is shown
below. Find the area of the doghouse floor.
1.5 m
2m
1.7 m
8. Draw a circle with a radius of 2.3 cm, and
then find its area.
9. MATHEMATICAL How does knowing the
R E F L E C T I O N area of a rectangle help
you find the areas of other figures?
Explain.
© 2010 College Board. All rights reserved.
14.3 in.
12 cm