Name Period Date GEOMETRY AND MEASUREMENT Student Pages for Packet 2: Circles GEO2.1 Circumference Use multiple representations to explore the relationship between the diameter and the circumference of a circle. Understand the concept of. Derive the circumference formula for circles. Solve circumference application problems. GEO2.2: Area of Circles Derive the area formula for circles. Solve application problems that involve areas of circles. GEO2 STUDENT PAGES GEO2.3 Vocabulary, Skill Builders, and Review 14 1 7 GEO2 SP
WORD BANK (GEO2) Word Definition Example or Picture center of a circle chord circle circumference diameter pi radius GEO2 SP0
2.1 Circumference Ready (Summary) We will explore the relationship between a circle's diameter and its circumference. We will learn about historical approximations for. We will derive the formula for the circumference of a circle and use it to solve problems. Use the figure to answer the questions below. CIRCUMFERENCE Go (Warmup) 1. Points on a circle are all equidistant from its. In the figure, this is represented by. 2. A line segment from the center of a circle to any point on the circle is called a. In the figure, this is represented by. Set (Goals) Use multiple representations to explore the relationship between the diameter and the circumference of a circle. Understand the concept of. Derive the circumference formula for circles. Solve circumference application problems. 3. A line segment with both endpoints on the circle is called a. In the figure, this is represented by. 4. A chord that goes through the center of the circle is called a. In the figure, this is represented by. A B M C D GEO2 SP1
2.1 Circumference MEASURING CIRCLES Use the table to record the diameter and circumference of objects measured in class. A. B. C. D. E. Object Diameter (d) Circumference (C) Circumference ( C) Diameter ( d) 1. The circumference of a circle is equal to about times the length of the diameter. 2. The diameter of a circle is times the radius. 3. Let d represent the length of the diameter. Let r represent the length of the radius. Let C represent the length of the circumference. a. Write an equation to describe the relationship between the circumference and the diameter. Use the symbol to represent is about equal to. b. Write an equation to describe the relationship between the diameter and the radius. c. Write an equation to describe the relationship between the circumference and the radius. Use the symbol to represent is about equal to. GEO2 SP2
2.1 Circumference A LITTLE HISTORY Many civilizations over the centuries have observed that the ratio of the circumference to the diameter of a circle is constant. For example, the Romans observed that the number of paces around the outer portion of their circular temples was about three times the number of paces through the center. In mathematics, the Greek letter (pronounced pi ) is used to represent this ratio. There is no fraction of integers that represents the exact ratio of the circumference to the diameter, or. Here are some approximations used by different civilizations over the ages. Decimal approximation for Fraction used as approximation for (to the nearest ten-thousandth) 1. Egyptian: 256 81 2. Greek: between 22 223 and 7 71 3. Hindu: 4. Roman: 5. Chinese: 6. Babylonian: 3,927 1,250 377 120 355 113 25 8 Today, the decimal approximation of, correct to five decimal places, is 7. Write this decimal approximation in words. 3.14159 8. Round this decimal approximation to the nearest ten-thousandth. 9. Which civilization had the best approximation for? GEO2 SP3
APPROXIMATING VALUES FOR 2.1 Circumference There is no fraction of integers that represents the exact value of. When solving algebra problems that involve, it is typical to keep the symbol in the problem and solution. However, in real-life calculations, it is often necessary to approximate an answer that involves. Compute two numerical approximations for each measurement that represents the circumference of a circle. (Hint: use your number sense to decide which approximations to compute.) Circumference of the circle 1. 7 2. 2.8 3. 100 4. 6 Approximate using 3 Approximate using 3.14 Approximate using 22 7 GEO2 SP4
2.1 Circumference The symbol for pi is. USE YOUR CIRCLE KNOWLEDGE 1 Common approximations for pi are,, and. Formulas for the circumference of a circle are and. Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. The plate problem: Calculate the circumference of a plate with a radius of 14 cm. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 2. The can problem: Calculate the diameter of the top of a soup can with a circumference of 32 cm. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question GEO2 SP5
2.1 Circumference USE YOUR CIRCLE KNOWLEDGE 1 (continued) Draw a picture, write the appropriate formula(s), and substitute to solve each problem. 3. The earth s orbit problem: The earth is about 93,000,000 miles from the sun, and the earth revolves around the sun one time per year. If the earth s orbit is approximately a circle, how far does the earth travel in one year? a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 4. The school track problem: A field at a local school is surrounded by a track. The straight-aways are each 425 feet. The distance across the field (top to bottom in the diagram) is 150 feet. Find the distance around the track. a. Label the figure Hint: The field is a rectangle with half-circles (or semicircles) at both ends. b. Write the appropriate formulas c. Substitute and solve d. Answer the question GEO2 SP6
2.2 Area of a Circle Ready (Summary) We will use our knowledge of the area of rectangles and circumference of circles to derive the area formula for a circle. We will use the formula to solve problems. AREA OF CIRCLES Go (Warmup) Set (Goals) Derive the area formula for circles. Solve application problems that involve areas of circles. Suppose the vertical and horizontal length between adjacent dots represents 4 feet. 1. Make a scale drawing of a 16 feet by 24 feet rectangle and find its area. 2. Make a scale drawing of a circle with a radius of 8 feet and find its circumference. GEO2 SP7
2.2 Area of a Circle DERIVING THE AREA OF A CIRCLE 1. What is the formula for the circumference of a circle in terms of its radius r? 2. Your teacher will give you a paper circle. Fold the circle in half. Fold it in half a second time. Fold it in half a third time. 3. Unfold your circle and cut along the folds to make 8 wedges. (You may want to try more than 8 wedges, like 16, but don t do less than 8.) 4. Arrange the wedges in a row. Alternate the tips up and down to form a shape that resembles a rectangle or parallelogram. Tape, glue stick, or sketch the shape here. Use your knowledge of the area of a rectangle and circumference of a circle to find the area of this shape. (Remember, it began as a circle and is becoming closer to resembling a rectangle. The more wedges you make, the closer it gets to becoming a rectangle.) 5. The width of the rectangle is of the circle. 6. The length of the rectangle is of the circle. 7. Write an equation for the approximate area of the rectangle. 8. Substitute the expression for the circumference of the circle from #1 above into your formula and simplify. 9. What is the formula for the area of a circle, in terms of the radius? GEO2 SP8
2.2 Area of a Circle USE YOUR CIRCLE KNOWLEDGE 2 Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. The dish problem: Find the area of a plate whose diameter is 12 inches. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 2. The water sprinkler problem: A revolving water sprinkler sprays water in a circular fashion to a distance of 20 feet in all directions. What area of grass can it cover? a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question GEO2 SP9
2.2 Area of a Circle USE YOUR CIRCLE KNOWLEDGE 2 (continued) Label the pictures, write the appropriate formulas, and substitute to solve each problem. 3. The shaded area problem: The largest possible circle is to be cut from a square board that is 56 inches on each side. What is the approximate area, in square inches, of the remaining board (shaded area)? a. Label the figure b. Write the appropriate formulas c. Substitute and solve d. Answer the question 4. The school track problem (revisited): A field at a local school is surrounded by a track. Each straightaway is 425 feet. The distance across the field (top to bottom in the diagram) is 150 feet. Find the area of the field. a. Label the figure straightaway b. Write the appropriate formulas c. Substitute and solve d. Answer the question GEO2 SP10
2.2 Area of a Circle THE GRADUATION CELEBRATION You are planning a graduation celebration for 75 students in your class. The room for the party has dimensions of 60 feet by 90 feet. You will construct a stage in the shape of a trapezoid with parallel edges that are 10 feet and 20 feet. The depth of the stage will be 15 feet. You want to put circular tables with diameters of 10 feet around the room, and you need to leave space for chairs and for people to walk around. Each table can seat up to 9 people. You also want to have a big square dance floor. 1. Use the dot paper on the next page to make a scale drawing of the room, and lay out where you will put the stage, dance floor, and tables. Label everything, and indicate how many people will sit at each table. (Hint: You may want to make a rough sketch of the room on dot or grid paper, and then transfer it onto this packet.) 2. What is the area of the stage? 3. How many tables do you need? What is the area of the space taken up by the tables? 4. What are the dimensions of your dance floor? GEO2 SP11
2.2 Area of a Circle SCALE DRAWING OF THE ROOM Use this page to make a rough sketch of the room. GEO2 SP12
2.2 Area of a Circle SCALE DRAWING OF THE ROOM Use this page to make the final draft of your scale drawing. GEO2 SP13
2.3 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (GEO2) Select the word from the word bank that best completes the sentence. 1. The of a circle is the distance around it. 2. A line segment with both endpoints on the circle is called a. 3. Points on a circle are equidistant from its. 4. A line segment from the center of a circle to any point on the circle is called a. 5. A is the set of all points in a plane that are a given distance (radius) from a given point (center). 6. A chord that goes through the center of a circle is called a. 7. 22 and 3.14 are approximations for. 7 Word Bank circle radius diameter chord circumference center of a circle pi GEO2 SP14
2.3 Vocabulary, Skill Builders, and Review Round to the nearest hundredths. SKILL BUILDER 1 1. 53.429 2. 0.6539 3. 100.985 Round to the nearest thousandths. 4. 0.35735 5. 510.7441 6. 1.0004 Write the following numbers in words: 7. 30.19 8. 85.073 Compute. 9. 1.11 2.0 12. 15.68 0.2 13. 0.3 45.6 10. 5.01 2.2 11. 0.9 6.2 14. 62.8 0.04 GEO2 SP15
2.3 Vocabulary, Skill Builders, and Review Compute. Simplify when possible. SKILL BUILDER 2 1. 3 5 4 7 2. 2 7 7 9 24 3 3. 8 5 1 1 5. 6 5 5 2 4. 12 9 4 8 6. 19 18 7 5 3 4 7. 2 8. 14 21 5 9 GEO2 SP16
2.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 1. Find the circumference of a circle whose radius is 14 inches. a. Write the solution as an exact answer. b. Write the solution as an approximation using 3. c. Write the solution as an approximation using 3.14. d. Write the solution as an approximation using 22 7. 2. Find the radius, circumference, and area of a circular table whose diameter measures 3.4 meters. Write your solutions as exact answers. Radius: Circumference: Area: 3. Pizza House sells 16-inch-diameter pizzas and 12-inch-diameter pizzas. How much more pizza would you get by ordering the larger pizza than the smaller one? GEO2 SP17
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2.3 Vocabulary, Skill Builders, and Review TEST PREPARATION (GEO2) Show your work on a separate sheet of paper and choose the best answer. 1. What is the circumference of a round table with a radius of 2.5 feet? A. C 5ft B. C 7.85 ft C. C 15.7 ft D. C 19.625 ft 2. What is the circumference of a circular rug with a radius of 1.5 yards? A. 9.42 yards B. 7.07 yards C. 4.71 yards D. 3 yards 3. The circumference of a bicycle wheel is 50.24 inches. What is the diameter? Use 3.14 for. A. 32 inches B. 16 inches C. 8 inches D. 4 inches 4. If a circle has a radius of 7 meters, what is the area of half of the circle? A. 76.9 m 2 B. 153.9 m 2 C. 44 m 2 D. 22 m 2 5. A pizza has a diameter of 18 inches. The pizza is cut into eight equal pieces. What is the area of each piece? Round to the nearest tenth. A. 254.3 sq. in B. 31.8 sq. in C. 56.5 sq. in D. 7 sq. in 6. What is the area of the circle rounded to the nearest tenth? 5.4 cm A. 8.5 cm B. 17 cm C. 22.9 cm D. 91.7 cm GEO2 SP19
2.3 Vocabulary, Skill Builders, and Review This page is left intentionally blank for notes. GEO2 SP20
2.3 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (GEO2) Show your work on a separate sheet of paper and write your answers on this page. 2.1 Circumference 1. Find the circumference of a circle whose radius is 8 meters. Approximate the measurement using 3 Approximate the measurement using 3.14 2. What is the circumference of a circle with a radius of 3 meters? Approximate the measurement using 22 7 3. The circumference of a compact disc is 28.26 centimeters. What is the diameter? 2.2 Area of Circles 4. Find the area of a circle with a radius of 12 km. 5. The area of a circle is 50.24 sq. cm. What is the radius of the circle? 6. Explain which has a greater area, a circle with radius 3 meters or a square with side length 3 meters. GEO2 SP21
Home-School Connection (GEO2) Here are some questions to review with your young mathematician. 1. The diameter of a nickel is 2 centimeters. What is the circumference? 2. What is the circumference of a circle with a radius of 3.5 meters? 3. What is the circumference of a plate with a radius of 7 in? What is the area of the plate? Use 22 to find the approximation of the measurement. 7 4. Which has a greater area, five circles, each with the radius 1 meter, or one circle with radius 5 meters. Parent (or Guardian) signature Selected California Mathematics Content Standards AF 6.3.1 Use variables in expressions describing geometric quantities (e.g. P = 2w 2l, A = 1 bh, C = d the 2 formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). MG 6.1.2 Know common estimates of (3.14; 22 ) and use these values to estimate and calculate the circumference 7 and the area of circles; compare with actual measurements. NS 7.1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. MG 7.2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. MG 7.2.2 Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects. FIRST PRINTING DO NOT DUPLICATE 2009 GEO2 SP22