5.3. Area of Polygons and Circles Play Area. My Notes ACTIVITY
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1 Area of Polygons and Circles SUGGESTED LEARNING STRATEGIES: Think/Pair/Share ACTIVITY 5.3 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 G 1. Look at the shape of each figure, and write the name of the playground equipment next to each letter. F A. B. C. D. E. F. G. Unit 5 Geometry 253
2 ACTIVITY 5.3 Area of Polygons and Circles SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/Peer Revision, Group Presentation 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 A B C D E F G Geometric Shape(s) 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 SpringBoard Mathematics with Meaning TM Level 1
3 Area of Polygons and Circles ACTIVITY 5.3 Two Congruent Parallelograms (Cut these out when you start Question 6.) Two Congruent Triangles (Cut these out when you start Question 10.) Two Congruent Trapezoids (Cut these out when you start Question 13.) Circle (Cut this out when you start Question 19.) Unit 5 Geometry 255
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5 Area of Polygons and Circles ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Look for a Pattern, Self Revision/Peer Revision, Debriefing 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 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.) 8. Find a relationship between the base, height, and area of a parallelogram. Describe that relationship using words, symbols, or both. 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 The height of a figure is always drawn perpendicular to its base. Perpendicular lines ( ) meet to form right angles. b. triangle c. pentagon Unit 5 Geometry 257
6 ACTIVITY 5.3 Area of Polygons and Circles 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. SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Quickwrite 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 A trapezoid is a quadrilateral with exactly one pair of parallel sides. 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. b 1 is read as b sub 1 and represents one base of the trapezoid. b 2 is read as b sub 2 and represents a second base of the trapezoid. 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. b On the inside of each figure label the 1 bases as b 1 and b 2 as shown at right. b 2 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. 258 SpringBoard Mathematics with Meaning TM Level 1
7 Area of Polygons and Circles ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Group Presentation, Self Revision/Peer Revision, Quickwrite 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? 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
8 ACTIVITY 5.3 Area of Polygons and Circles SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Guess and Check, Group Presentation, Debriefing 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? 23. Using words, symbols, or both, describe how you can now find the area of the circle. 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 Meaning TM Level 1
9 Area of Polygons and Circles ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Group Presentation 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 3 ft 2 ft 2 ft 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. e. Figure B Recall that we can approximate π as either 3.14 or ft Unit 5 Geometry 261
10 ACTIVITY 5.3 Area of Polygons and Circles SUGGESTED LEARNING STRATEGIES: Think/Pair/Share 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. Find the area of each figure. Remember to label your answer in. 5. Use π 22 to find the area of the circle Find the area. 14 in. 7 in in. 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 12 cm 8 cm 15.4 cm 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 8. Draw a circle with a radius of 2.3 cm, and then find its area. 9. MATHEMATICAL REFLECTION 14.3 in. 2 m 1.7 m How does knowing the area of a rectangle help you find the areas of other figures? Explain. 262 SpringBoard Mathematics with Meaning TM Level 1
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