DIAMETER d flower pot 12.5 in. 4.0 in. penny 6.0 cm 1.9 cm trash can lid 75.5 in in. patio table in in. coffee mug 26.2 cm 8.

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1 1. There is a relationship between the diameter and circumference of any circle. Study the following diagrams and describe what you notice about the circumference and diameter of each circle. 2. The following chart shows the measurements of several round objects. For each item take the circumference and divide by the diameter. Round your answers to the nearest one tenth of a unit. What do the results tell you about the circumference and diameter of any circle? OBJECT CIRCUMFERENCE C DIAMETER d flower pot 12.5 in. 4.0 in. penny 6.0 cm 1.9 cm trash can lid 75.5 in in. patio table in in. coffee mug 26.2 cm 8.6 cm C d RATIO OF C TO d 3. Carlos has a friend who is stumped trying to find how much edging he needs for a circular garden. The diameter of the garden is 16 ft. Use what you know about circumference and diameter to estimate the circumference of the garden. 4. The formula for circumference of a circle is C = d. Use this formula to find the circumference of the garden. 1

2 5. The radius square is a square whose side is the length of the radius of a circle. There is a relationship between the radius square and area of any circle. Study the diagrams below and estimate the number of radius squares it takes to cover the circle? 6. Carlos friend also wants to find the area of his circular garden. The radius of the garden is 8 ft. Use what you learned in problem 4 to estimate the area of the garden. 7. The formula for area of a circle is A = r 2. Use this formula to find the area of the garden. 2

3 After watching the What is? video, make sense of the mathematics by taking a closer look at the problem situation and solutions. Use the questions and comments in bold to help you make sense of. There is a relationship between the diameter and circumference of any circle. Study the following diagrams and describe what you notice about the circumference and diameter of each circle. No matter how large or small the circle it always takes a little more than three diameters to equal the circumference. This is true of the top of a circular flowerpot and the top of the mug. It is also true of very small circles like a coin and very large circles like the center circle of a soccer field. The following chart shows the measurements of several round objects. For each item, divide the circumference by the diameter. Round your answers to the nearest one tenth of a unit. What do you notice about the results? OBJECT CIRCUMFERENCE C DIAMETER d flower pot 12.5 in. 4.0 in. penny 6.0 cm 1.9 cm trash can lid 75.5 in in. patio table in in. coffee mug 26.2 cm 8.6 cm C d RATIO OF C TO d When circumference is divided by diameter, the ratios are always a little more than three. If we could be exact in the measurements, we would get the same number every time and that number is pi ( ). 3

4 OBJECT CIRCUMFERENCE C DIAMETER d C d RATIO OF C TO d flower pot 12.5 in. 4.0 in. 3.1 penny 6.0 cm 1.9 cm 3.2 trash can lid 75.5 in in. 3.2 patio table in in. 3.1 coffee mug 26.2 cm 8.6 cm 3.0 Pi ( ) is a Greek symbol for an irrational number. An irrational number goes on forever without ever repeating. Mathematicians have calculated to over a million decimal places. How does this relationship relate to the formula, C= d? If the circumference of any circle divided by the diameter of that circle equals, then the circumference is equal to times diameter. If C d =, then C = d. Carlos has a friend who is stumped trying to find the size of a circular garden and how much edging it will take. The diameter of the garden is 16 ft. Knowing the circumference will allow the friend to find the amount of edging necessary for the garden. What is the circumference of the garden? The diameter of the garden is 16 ft., so the circumference is 16. To find the approximate circumference, substitute 3.14 for. Since the diameter of the circle was measured to the nearest foot, it is more accurate to report the circumference of the circle to the nearest foot. Therefore, the circumference of the garden is about 50 ft. 4

5 There is also a relationship between the area and radius of any circle. What is the radius of a circle? The radius of a circle is the distance from a point on the circle to the center of the circle. What is the radius square? The radius square is a square whose side is the length of the radius. How many radius squares will it take to cover the circle? 5

6 One radius square fits easily. You can cut up a second radius square and make it fit. If you cut up a third radius square even more, it also fits. Just a little bit of the circle is not covered, so it takes a little more than three radius squares to cover the circle. The area of any circle is approximately three times its radius squared or exactly times the radius squared (Area of a circle = r 2 ). 6

7 You already found the circumference of the circular garden. Now find the area. The radius of the garden is 8 ft. The area is 8 2 or about 3.14 x 64 which is approximately sq. ft. Since the radius was measured to the nearest foot, it would be more accurate to report the area as approximately 201 sq. ft. There is a relationship between the diameter and the circumference of any circle and there is a relationship between the radius and area of any circle. No matter how large or small the circle, it always takes a little more than three diameters, or exactly diameters, to equal the circumference. No matter how large or small the circle, it always takes a little more than three radius squares, or exactly radius squares, to equal the area. 7

8 1. The following chart lists the circumference and diameter of several circular objects. Complete the chart using the following directions. a. Find the ratio of the circumference to the diameter of each object using division (C d). Round each answer to two decimal places and record it in the last column of the chart. b. Find three more circular objects and record each object in the blank rows at the bottom of the chart. Measure the circumference and diameter of each object you found and record the measurements in the appropriate column. Do not measure the diameter and compute the circumference. If you do not have a tape measure, you may want to use a string to help you measure the circumference. c. Find the ratio of the circumference to the diameter using division (C d) and record the results in the last column. d. Compare your results to the results found in the video. Object Circumference Diameter Ratio of C to d basketball hoop 1.41 m 0.45 m penny mm mm soccer field center circle m m 2. Emily owns a circular trampoline that has a diameter of 12 feet. She needs to replace the safety net around the trampoline. How long must the netting be to completely surround the trampoline? 8

9 3. While attending summer camp, Hamish and Braden took a walk around a small circular lake. The distance around the lake was 5.5 km. The next day they swam across the lake. If they swam across the widest part of the lake, how far did they swim? 4. College Town Pizza sells two sizes of pizzas, individual and family size. The individual pizza has a diameter of 8 inches and the family size pizza has a diameter of 16 inches. Find the area of each pizza. 5. A garden sprinkler advertises that it waters a circular region covering 336 to 400 square feet. Anton reseeded a circular portion of his lawn and needs to water it. The diameter of this newly seeded lawn is 20 feet. If the advertised sprinkler is placed in the center of the seeded lawn, will Anton have to move the sprinkler in order to water the entire section of new grass? 9

10 1. The following chart lists the circumference and diameter of several circular objects. Complete the chart using the following directions. a. Find the ratio of the circumference to the diameter of each object using division (C d). Round each answer to two decimal places and record it in the last column of the chart. b. Find three more circular objects and record each object in the blank rows at the bottom of the chart. Measure the circumference and diameter of each object you found and record the measurements in the appropriate column. Do not measure the diameter and compute the circumference. If you do not have a tape measure, you may want to use a string to help you measure the circumference. c. Find the ratio of the circumference to the diameter using division (C d) and record the results in the last column. d. Compare your results to the results found in the video. Object Circumference Diameter Ratio of C to d basketball hoop 1.41 m 0.45 m 3.13 penny mm mm 3.14 soccer field center circle m m 3.14 Measurement is never exact and always involves some error. The ratio of circumference to diameter should be a little greater than three for any circular object you measure. Depending on the accuracy of your measurements, your results may vary. 2. Emily owns a circular trampoline that has a diameter of 12 feet. She needs to replace the safety net around the trampoline. How long must the netting be to completely surround the trampoline? The trampoline has a diameter of 12 feet, so the netting will need to be a little more than 3 times 12, or 36 feet. If you use 3.14 to approximate pi, the answer will be about 38 feet of netting (12 x 3.14 = 37.68). 10

11 3. While attending summer camp, Hamish and Braden took a walk around a small circular lake. The distance around the lake was 5.5 km. The next day they swam across the lake. If they swam across the widest part of the lake, how far did they swim? The distance around the lake is the circumference. The circumference is equal to pi times the diameter, so 5.5 km = d. If you use 3.14 to approximate pi, then d , or 1.8 km. 4. College Town Pizza sells two sizes of pizzas, individual and family size. The individual pizza has a diameter of 8 inches and the family size pizza has a diameter of 16 inches. Find the area of each pizza. The diameter of the individual pizza is 8 inches, so the radius is 4 inches. To find the area of a circle, multiply pi times the radius squared. If you use 3.14 for pi, the area will be approximately 3.14 x 4 2, or 50 square inches. The diameter of the family size pizza is 16 inches, so the radius is 8 inches. If you use 3.14 for pi, the area will be approximately 3.14 x 8 2, or 201 square inches. 5. A garden sprinkler advertises that it waters a circular region covering 336 to 400 square feet. Anton reseeded a circular portion of his lawn and needs to water it. The diameter of this newly seeded lawn is 20 feet. If the advertised sprinkler is placed in the center of the seeded lawn, will Anton have to move the sprinkler in order to water the entire section of new grass? If the diameter of the newly seeded lawn is 20 feet, then the radius is 10 feet. A circle whose radius is 10 feet has an area of approximately 314 square feet (3.14 x 10 2 ). Since the sprinkler covers between 336 and 400 square feet, the entire seeded area can be watered without moving the sprinkler. 11

12 1. Below is a table listing the circumference and diameter of several objects. Graph these values on the grid using the horizontal axis for diameter and the vertical axis for circumference. Object Circumference Diameter CD case 40.1 cm 12.5 cm Coffee mug 25.3 cm 8.0 cm Salad plate 55.8 cm 17.9 cm Pencil top 2.2 cm 0.7 cm Penny 6.3 cm 2.0 cm Quarter 7.7 cm 2.4 cm Lotion bottle 11.4 cm 3.4 cm Water bottle 22.6 cm 7.0 cm Circumference in cm diameter in cm Describe your graph. What does your graph tell you about the relationship between circumference and diameter? 12

13 2. Mr. Garcia gave the following problem to students in his math class. A decorative fountain pool at school has a diameter of 28 feet. What is the approximate circumference of the pool? What is the approximate area?! Mr. Garcia s students gave the following answers. Explain whether or not the answers are reasonable estimates. Student A: C = 90 ft. A = 2700 sq. ft. Student B: C = 84 ft. A = 600 sq. ft. Student C: C = 150 ft. A = 600 ft. Student D: C = ft. A = sq. ft. 13

14 1. Below is a table listing the circumference and diameter of several objects. Graph these values on the grid using the horizontal axis for diameter and the vertical axis for circumference. Object Circumference Diameter CD case 40.1 cm 12.5 cm Coffee mug 25.3 cm 8.0 cm Salad plate 55.8 cm 17.9 cm Pencil top 2.2 cm 0.7 cm Penny 6.3 cm 2.0 cm Quarter 7.7 cm 2.4 cm Lotion bottle 11.4 cm 3.4 cm Water bottle 22.6 cm 7.0 cm Circumference in cm diameter in cm Describe your graph. What does your graph tell you about the relationship between circumference and diameter? The points on the graph look like they form a straight line. The slope of the line is a little more than three. That tells us that the circumference of a circle is a little more than three diameters. If measurements could be made without error, the line would have a slope equal to pi. 14

15 2. Mr. Garcia gave the following problem to students in his math class. A decorative fountain pool at school has a diameter of 28 feet. What is the approximate circumference of the pool? What is the approximate area?! Mr. Garcia s students gave the following answers. Explain whether or not the answers are reasonable estimates. Student A: C = 90 ft. A = 2700 sq. ft. This answer for the circumference is reasonable. The formula of circumference is C = d. The diameter is close to 30 and is close to three, so the circumference is close to 3 x 30, or 90 ft. The answer for the area is not reasonable. (A = r 2 ) It appears that the student squared 30, the approximate diameter (30 2 x 3), instead of the radius (14 2 x 3). Student B: C = 84 ft. A = 600 sq. ft. This answer for circumference is reasonable. The formula for circumference is C = d. The diameter is 28 feet and is close to three, so the circumference is close to 3 x 28, or 84 ft. This answer for area is also reasonable. (A = r 2 ) The radius is half the diameter or 14 ft. The radius squared is 196 or about 200 (200 x 3 = 600 sq. ft.). Student C: C = 150 ft. A = 600 ft. This answer for circumference is not reasonable. The formula for circumference is C = d. It appears that this student doubled the diameter (about 50) and then multiplied by 3 to get 150. This answer for area is not reasonable because the label is incorrect. The area is approximately 600 sq. ft. rather than 600 ft. You cannot measure area with a linear unit. Student D: C = ft. A = sq. ft. The student should indicate that both answers are approximations rather than exact answers. Since pi is an irrational number, any measurement found by substituting 3.14 for pi is an approximation (C 28 x 3.14 and A 14 2 x 3.14). 15

16 Scene Full Transcript 1 Carlos: Hey, everybody! It s Carlos. My friend Bre is into architecture and landscape design. When her teacher required everyone in the class to do a job shadow on a career that they were interested in, Bre knew just where to go, Larry s Landscaping. 2 Voice- Over Carlos: Her week is off to a good start; she s learning all aspects of the business. She even came up with an awesome design of her own. The only problem is that she didn t calculate how much math goes into landscaping. She s stumped trying to find out the area of this circular garden and how much edging it will take. 3 Carlos: Looks like we need a lesson on understanding pi. Grab a shovel and let s dig into another Problem Solved. 4 Carlos: Here s Bre s design. First, we need to understand that there is a relationship between the diameter and the circumference of any circle. Here s how I explained it to Bre. 5 Voice- Over Carlos: 6 Voice- Over Carlos: 7 Voice- Over Carlos: I asked her to compare the circumference and diameter of several circular items around the shop, everything from flowerpots to coffee cups. Watch what happens. No matter how large or small the circle, it always takes a little more than three diameters to equal the circumference. If you don t believe me, try it for yourself. It works. Now, let s measure the circumference and the diameter of each of these objects and record the measurements in the chart. Then, we divide the measure of the circumference by the measure of the diameter. Look at these ratios! You always get a little more than three. You can even use different units of measurement. It will always be about three. 8 Carlos: And, if we could be exact, we would get the same number every time, and that number is pi. Pi is a Greek symbol for an irrational number. That means that it goes on forever without ever repeating. Mathematicians, with the help of computers, have calculated pi to over a million decimal places. Whoa! 9 Voice- Over Carlos: Let s look at this circular garden. If the circumference divided by diameter equals pi, then circumference equals pi times diameter. Knowing the circumference will tell Bre how much edging she needs. The diameter of this circle is 16 feet, so the circumference is about feet. Can you 16

17 believe how easy that is? 10 Voice- Over Carlos: Now, Bre needs to determine the area of the circle for planting. Pi is also used to find the area of a circle. You can find the area of a circle by using the formula r 2, but do you really understand why? R is the radius, but what is the radius squared? Here is the radius, and here is the radius squared. Let s cover the circle with radius squares. How many do you think it will take? One fits easily. We can cut up a second radius square to make it fit. We need to cut the third radius square up even more, but it also fits. Just a little bit of our circle is not covered, so it takes a little more than three radius squares to cover our circle, just like it took a little more than three diameters to equal the circumference. 11 Carlos: The important relationship to remember is that the area of any circle is approximately 3 times its radius squared or exactly pi times the radius squared. 12 Voice- Over Carlos: Back to our garden the radius is 8 feet, so the area is pi times 8 squared. That s pi times 64 or approximately square feet. 13 Carlos: It works every time, as designed. Bre here is set on her new career path, and now you know the meaning of pi. Problem Solved. Oh, I love this. 17