Pi: The Ultimate Ratio
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- Donald Rudolph Barrett
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1 Pi: The Ultimate Ratio Exploring the Ratio of Circle Circumference to Diameter 1 WARM UP Scale up or down to determine an equivalent ratio miles 3 hours 5? 1 hour 2. $750 4 days in. 1 ft 48 oz 3 lb 5 5 5? 1 day? 5 ft? 1 lb LEARNING GOALS Identify pi (π) as the ratio of the circumference of a circle to its diameter. Construct circles using a compass and identify various parts of circles. Know and write the formula for the circumference of a circle, and use the formula to solve problems. KEY TERMS congruent circle radius diameter circumference pi You have learned about ratios. How can you use ratios to analyze the properties of geometric figures such as circles? LESSON 1: Pi: The Ultimate Ratio M1-7
2 Getting Started Across and Around A circle is shown with a point drawn at the center of the circle. The name of the point is O, so let s call this Circle O. O Be sure to include units when you record your measurements. 1. Analyze the distance around the circle. a. Use a string and a centimeter ruler to determine the distance around the circle. b. How does your measurement compare to your classmates measurements? Summarize the similarities and differences. 2. Draw a line from a point on the circle to the center of the circle, point O. a. Measure your line using your centimeter ruler. b. How does your measurement compare to your classmates measurements? Summarize the similarities and differences. M1-8 TOPIC 1: Circles and Ratio
3 ACTIVITY 1.1 Analyzing the Parts of a Circle Everyone can identify a circle when they see it, but defining a circle is a bit harder. Can you define a circle without using the word round? Investigating how a circle is formed will help you mathematically define a circle. 1. Follow the given steps to investigate how a circle is formed. Step 1: In the space provided, draw a point and label the point A. Step 2: Use a centimeter ruler to locate and draw a second point that is exactly 5 cm from point A. Label this point B. Step 3: Locate a third point that is exactly 5 cm from point A. Label this point C. 2. How many other points could be located exactly 5 cm from point A? How would you describe this collection of points in relation to point A? Step 4: Repeat this process until you have drawn at least ten distinct points that are each exactly 5 cm from point A. 3. Define the term circle without using the word round. LESSON 1: Pi: The Ultimate Ratio M1-9
4 A circle is a collection of points on the same plane equidistant from the same point. The center of a circle is the point from which all points on the circle are equidistant. Circles are named by their center point. F E A C B H D G 4. Use the circle shown to answer each question. a. Name the circle. The radius of a circle is a line segment formed by connecting a point on the circle and the center of the circle. The distance across a circle through the center is the diameter of the circle. The diameter of a circle is a line segment formed by connecting two points on the circle such that the line segment passes through the center point. The distance around a circle is called the circumference of the circle. b. Identify a radius of the circle. The plural of radius is radii. c. Identify a diameter of the circle. d. Are all radii of this circle the same length? Explain your reasoning. 5. What is the relationship between the length of a radius and the length of a diameter? M1-10 TOPIC 1: Circles and Ratio
5 ACTIVITY 1.2 Measuring the Distance Around a Circle Let s explore circles. Use circles A, B, D, E, and O provided at the end of the lesson. Circle O is the same as the circle from the activity Across and Around. 1. Use a string and a centimeter ruler to measure the distance from a point on the circle to the center and the distance around each circle. Record your measurements in the table. In the last column, write the ratio of Circumference : Diameter in fractional form. Circle Circumference Radius Diameter Circumference Diameter Circle A Circle B Circle O Circle D Circle E 2. Average the ratios recorded for Circumference. What is the Diameter approximate ratio for the circumference to the diameter for the set of circles? Write the approximate ratio as a fraction and as a decimal. 3. How does your answer to Question 2 compare to your classmates answers? 4. Average all of your classmates answers to Question 3. Write the approximate ratio of circumference to the diameter as a fraction and as a decimal. LESSON 1: Pi: The Ultimate Ratio M1-11
6 ACTIVITY 1.3 The Circumference Formula NOTES The number pi (π) is the ratio of the circumference of a circle to its circumference of a circle diameter. That is pi 5, or π 5 C, where C is diameter of a circle d the circumference of the circle, and d is the diameter of the circle. The number π has an infinite number of decimal digits that never repeat. Some approximations used for the value π are 3.14 and Use this information to write a formula for the circumference of a circle, where d represents the diameter of a circle and C represents the circumference of a circle. 2. Rewrite the formula for the circumference of a circle, where r represents the radius of a circle and C represents the circumference of a circle. 3. Use different representations for π to calculate the circumference of a circle. a. Calculate the circumference of a circle with a diameter of 4.5 centimeters and a circle with a radius of 6 inches. Round your answer to the nearest ten-thousandths, if necessary. Value for π d centimeters r 5 6 inches π Use the π Key on a Calculator Use 3.14 for π Use 22 7 for π M1-12 TOPIC 1: Circles and Ratio
7 b. Compare your circumference calculations. How do the different values of π affect your calculations? When you use 3.14 for pi, your answers are approximations. But an answer like 12π is exact. 4. Use the circumference of a circle formula to determine each unknown. Use 3.14 for π. a. Compute the diameter of the circle with a circumference of feet. b. Compute the radius of the circle with a circumference of millimeters. 5. What is the minimum amount of information needed to compute the circumference of a circle? LESSON 1: Pi: The Ultimate Ratio M1-13
8 NOTES TALK the TALK Twice Use what you have learned to compare circles by their characteristics. 1. Draw each circle. a. radius length of 3 centimeters b. diameter length of 3 centimeters 2. Describe the similarities and differences between your two circles. 3. Describe the relationship between the circumferences of the two circles. 4. Describe the circumference-to-diameter ratio of all circles. M1-14 TOPIC 1: Circles and Ratio
9 Measuring the Distance Around a Circle A B O D E LESSON 1: Pi: The Ultimate Ratio M1-15
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WARM-UPS Part I. Unit 5 - Circles DEGREES cm A. C =20.75 cm REMINDER: REMINDER: Circumference = 2πr = πd. 1. Find the circumference of :
WARM-UPS Part I REMINDER: Circumference = 2πr = πd REMINDER: DEGREES 360 ARC CIRCUMFERE NCE 1. Find the circumference of : C 4.37 cm 6.24 cm A 85 B C= C= C= 2. Find the radius of : Find the diameter of:
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. D - To evaluate the expression, we can regroup the numbers and the powers of ten, multiply, and adjust the decimal and exponent to put the answer in correct scientific notation format: 5 0 0 7 = 5 0
What You ll Learn. or irrational. New Vocabulary perfect square, square root, irrational numbers, real numbers. Why Learn This?
-. Plan - Exploring Square Roots and Irrational Numbers Objective To find and estimate square roots and to classify numbers as rational or irrational Examples Finding Square Roots of Perfect Squares Estimating
CHM101 Lab Measurements and Conversions Grading Rubric
CHM101 Lab Measurements and Conversions Grading Rubric Name Team Name Criteria Points possible Points earned Lab Performance Printed lab handout and rubric was brought to lab 3 Safety and proper waste
Name two radii in Circle E.
A C E B D Name two radii in Circle E. Unit 4: Prerequisite Terms A C E B D ECandED Unit 4: Prerequisite Terms A C E B D Name all chords in Circle E. Unit 4: Prerequisite Terms A C E B D AD, CD, AB Unit