The Circumference / Diameter Ratio JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Recall that: Now: The distance around a circle is called the circumference. The distance around a polygon is called the perimeter. Radius Diameter Center Definitions JRLeon Geometry Chapter 6.5 HGSH

The Circumference / Diameter Ratio Which is greater, the length of three tennis-balls or the circumference of one tennis-ball? SURPRISED? Wrap a string around the one ball to measure its circumference, then compare this measurement with the length of the three tennis-balls. JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Now that you actually compared the measurements, you discovered that the circumference of the tennis-ball is greater than three diameters of the ball. In this lesson you are going to discover the relationship between the diameter and the circumference of every circle. Once you know this relationship, you can measure a circle’s diameter and calculate its circumference. Let’s go investigate this relationship! JRLeon Geometry Chapter 6.5 - 6.6 HGSH

The Circumference / Diameter Ratio Object Circumference (C) Diameter (d) Ratio 𝑪 𝒅 1. 2. 3. Compare your average with the averages of other groups. Are the 𝐶 𝑑 ratios close? By now you should be convinced that the ratio 𝐶 𝑑 is very close to 3 for every circle. We define  as the ratio 𝐶 𝑑 and the ratio is approximated as 3.14. If you solve this formula for C, you get a formula for the circumference of a circle in terms of the diameter, d ( C = d ). The diameter is twice the radius (d =2r), so you can also get a formula for the circumference in terms of the radius, r ( C = 2r ) . If you’re asked for an exact answer instead of an approximation, state your answer in terms of  JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Circumference Conjecture If C is the circumference and d is the diameter of a circle, then there is a number such that C = __. If d = 2r where r is the radius, then C = ___. C-65 d 2r A Little History The first calculation of the circumference to diameter ratio, known as pi ( ), was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes showed the ratio to be between 3 1 7 and 3 10 71 Mathematicians in ancient Egypt used ( 4 3 ) 4 as their approximation of circumference to diameter. Early Chinese and Hindu mathematicians used 10. By 408 A.D, Chinese mathematicians were using 355 113 . Today, computers have calculated approximations of  to billions of decimal places! JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Example 1: If a circle has diameter 3.0 meters, what is the circumference? Use a calculator and state your answer to the nearest 0.1 meter. Solution: C = d The Circumference Formula C = (3.0) Substitute d with 3.0 In terms of , the answer is 3 . The circumference is approximately 9.4 meters JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Example 2: If a circle has circumference 12 meters, what is the radius? Solution: C = 2r The Circumference Formula 12 = 2r Substitute the value of C 6= r Substitute the value of C So the radius is 6 meters. JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Example 3 – Applications based on  : Jules Verne’s “Around the world in 80 days” If the diameter of Earth is 8000 miles, find the average speed in miles per hour Phileas Fogg needs to circumnavigate Earth about the equator in 80 days. Solution: C = d The Circumference Formula C = (8000) Substitute the value of d, where d = 8000 miles C =  (8000) C  3.14(8000) Substitute the value of   3.14 C  25,133 miles So, Phileas must travel 25,133 miles in 80 days. To find the speed v in mi/h, you need to divide distance by time and convert days into hours. d = rt rate x time = distance 25,133 = r(80) rate x time = distance where d=25,133 miles and t=80 days 𝟐𝟓,𝟏𝟑𝟑 𝒎𝒊𝒍𝒆𝒔 𝟖𝟎 𝒅𝒂𝒚𝒔 x 𝟏 𝒅𝒂𝒚 𝟐𝟒 𝒉𝒐𝒖𝒓𝒔 = 13 mi./hr. JRLeon Geometry Chapter 6.5 – 6.6 HGSH

The Circumference / Diameter Ratio Classwork / Homework: LESSON 6.5 : Pages 337-339, problems 1 through 15 LESSON 6.6 : Page 342, problems 1 through 4 . JRLeon Geometry Chapter 6.5 HGSH