Private: Learning Math: Measurement
Circles and Pi (π) Part A: Circles and Circumference (60 minutes)
Session 7, Part A
In This Part
- Ratio of Circumference and Diameter
- Pi (π)
Perimeter — or distance around — is a measurable property of simple, closed curves and shapes. When the figure is a circle, we use the term circumference instead of perimeter. Because the perimeter of an object is a length, we need to measure using units of length such as centimeters, decimeters, meters, inches, feet, etc.
The circumferences of circular objects can be difficult to estimate. Let’s use a bicycle wheel as an example. How long is its circumference? Use masking tape to make a mark on the floor or table to indicate the starting point. Estimate the distance of one rotation of the wheel or bowl rim, namely the circumference, by placing another piece of tape on the floor or table.
Roll the wheel to find the actual circumference. Was your estimate too short or too long?
There is a relationship between the circumference and diameter of a circle, which we will explore here in a number of ways. A diameter is a chord — a line segment joining two points on the arc of a circle — that passes through the center of the circle. Diameter also refers to the distance between two points on the circle, measured through the center. Let’s first look for patterns in the measurements of circles.
The three designs below show a circle between a regular hexagon and a square.
Print a PDF image (be sure to print this document full scale) of the designs to work on Problems A1-A3.
Use the designs to fill in the table below. For the circle, use string to approximate the circumference.
|Diameter of the Circle
|Perimeter of the Hexagon
|Perimeter of the Square
|Approximate Circumference of the Circle
Look closely at the three designs. What patterns do you see in their measurements?
- For each design, how does the diameter of the circle compare to the perimeters of the square and the hexagon?
- For each design, how does the approximate circumference compare to the perimeters of the square and the hexagon?
- The circumference of any of these circles is about how many times more than its diameter? If a circle had a diameter of 7 cm, what prediction would you make for the length of its circumference? Why?
Ratio of Circumference and Diameter
You can measure circular objects to verify the pattern you’ve seen. Choose four or five different circular objects to measure.
- For each object, estimate the circumference. Then measure the circumference and the diameter in centimeters to the nearest tenth (e.g., millimeters). Use string or a tape measure. Record your data in the table.
- Examine the table. What do you notice about the ratio of C to d? Based on these data, what is the relationship between the diameter and circumference of a circle?
- Enter the values from the table for diameter and circumference into a graphing program in your computer or into a table in your graphing calculator to make a scatter plot. Use the horizontal axis (x) for diameter and the vertical axis (y) for circumference. Graph the points. What patterns do you see in the graphical data?
- What information does a graph of these data provide?
A straight line indicates that the data are increasing by a constant amount. What is the constant in this case?
Find the mean of the data in the C/d column. Why find the mean? Does the mean approximate π?
A mean is an average, or a sum of all the data values divided by the total number of data values.
By now you have seen that all circles have one trait in common: The ratio of circumference to diameter is a constant value, π, which is a little more than 3. Pi is an irrational number that is represented by the symbol π.Its decimal part continues forever without repeating. As of 1997, π had been extended to 51 billion decimal places (using a computer)! Your calculator has a special key for π, but this is only an approximate value.
|Video SegmentIn this video segment, participants investigate the relationship of circumference and diameter using different circular objects. They collect the data by measuring, and then make observations about their findings.
Were you surprised to find out that , which is an irrational number, can be expressed as a constant ratio of two numbers, namely the diameter and circumference of any circle?
You can find this segment on the session video approximately 9 minutes and 16 seconds after the Annenberg Media logo.
The symbol r represents the radius of a circle. Explain why C = π • 2r is a valid formula for the circumference of a circle.
An irrational number cannot be written as a quotient of any two whole numbers. Yet we sometimes see π written as 22/7 or 3.14. Explain what the reason for this may be.
Since π is an irrational number, can both the circumference and the diameter be rational numbers? Can one of them be rational? Explain using examples.
When mathematicians are asked to determine the circumference of a circle, say with a diameter of 4 cm, they often write the following:
C = π • d = π • 4
In other words, the circumference of the circle is 4 cm. Why do you think they record the answer in this manner? Why not use the key on the calculator to find a numerical value for the circumference?
Is the value for π given on a calculator an approximation or an exact amount?
Note that we have worked with two forms of the standard equation that shows the relationship between circumference and diameter:
C = π • d π = C/d
Since π is an irrational number, the exact circumference can only be expressed using the symbol for π. Sometimes, however, we want to solve a real problem and find an approximate value for a circumference. In that case, we must use one of the approximations for π. Inexactness may also occur when determining a numerical value for circumference (or diameter) because of measurement error.
If you do not have a bicycle wheel available, you can start this section by estimating the circumference of different circular objects, such as the rim of a large bowl or can. Cut a piece of string the length of the estimate and then compare the estimate to the actual circumference. Most people grossly underestimate circumference.
It is important to measure as accurately as possible to avoid measurement errors.
You can again measure the diameter of the bicycle wheel and then use that to estimate the circumference based on observations you make in this problem.
Entering coordinates (diameter and circumference) for more than five objects might show a better approximation of a line. You can also find the line of best fit to see the patterns. To learn more about scatter plots and the line of best fit, go to Learning Math: Data Analysis, Statistics, and Probability, and find Session 7, Part A and Part D respectively.
To learn more about mean, go to Learning Math: Data Analysis, Statistics, and Probability, Session 5.
Here is the completed table:
The measurements stay in scale. In all three, the diagonal of the hexagon is twice the length of the hexagon’s side. Also, as we move from one design to the next, the length of each side of the inscribed hexagon increases by 1; the length of each side of the inscribed hexagon is equal to the radius of the circle (as shown by the inscribed equilateral triangles):
The length of each side of the square is the same as the diameter of the circle inscribed within; the ratio of the length of the diameter of the circle to the length of one side of the hexagon is 2/1 for all three designs.
- The perimeter of the hexagon is three times the diameter of the circle, and the perimeter of the square is four times the diameter of the circle.
- The circumference of the circle is between these two values, and closer to the hexagon’s perimeter.
- The circumference appears to be between 3.1 and 3.2 times larger than the diameter. If a circle had a diameter of 7 cm, you might predict its circumference to be somewhere near 22 cm. As we explore this further, we will see that the relationship between the circumference and diameter is a constant value.
- Answers will vary.
- The measured ratio of C/d should be approximately the same for all circular objects. It seems that the relationship between diameter and circumference is linear, and there is some number k so that C = k • d for every circle’s circumference and diameter. For now, we can say that this number is just slightly larger than 3.
- Answers will vary. The points should roughly form a straight line. If you were to place a line of best fit onto your scatter plot, the line would be y = 3.14 • x.Notice that the ratio C/d is about 3.14 no matter what the size of the circle. This is called a constant ratio since the value is constant, regardless of the circle. Constant change is represented by a straight-line graph and is sometimes referred to as a linear relationship. If the ratio between circumference and diameter differed for every circle, the graph would not be a straight line.
- This suggests that the diameter and circumference are in direct variation; that is, the circumference is a direct multiple of the diameter. Note that since you measured the circumference and diameter, there are likely to be measurement errors which will affect the graphed data.
Answers will vary, but should be close to 3.14 (an approximation for π). Finding the mean minimizes any measurement errors in the calculations of Problem A5.
We’ve seen that π = C/d, where C is the circumference and d is the diameter of a circle. We can multiply both sides of the equation by d to get a new equation as C = πd. Because the diameter of a circle is always twice its radius, we can write the new equation as C = π • 2r, which is what we wanted.
These forms make calculations involving π easier by using an approximation. In cases where there may already be measurement error, it doesn’t make sense to use an overly accurate version of π. Fractions like 22/7 or decimals like 3.14 do the job nicely in different situations.
In practical terms, it is impossible to buy, for example, a length of fencing that measures 4π. In applications, we often want an approximation that we can measure and work with. In mathematics problems, however, it is almost always preferable to use the symbol π.
One or the other may be rational, but not both. If they were both rational, their ratio (which is π) would also have to be rational, which it is not. A circle may have a diameter of exactly 12 cm with an irrational circumference, or a circumference of exactly 100 m with an irrational diameter.
They can, however, both be irrational.
This answer provides the only way to write the answer exactly. Additionally, it is easier to perform arithmetic with 4π than with a decimal approximation of it. An approximation may be substituted later if needed.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.