CIRCUMFERENCE OF A CIRCLE
INTRODUCTION
The objective for this lesson on Circumference of a Circle is, the student will explore the circumference of a circle and apply the formula to solve mathematical and real-world problems.
The skills students should have in order to help them in this lesson include, perimeter of a square.
We will have three essential questions that will be guiding our lesson. Number One, explain the relationship between the radius and diameter of a circle. Number Two explain the relationship between the diameter and circumference of a circle. And Number Three, why is the circumference of a circle an approximate value? Justify your answer.
Begin by completing the warm-up on perimeter of triangles and rectangles to prepare for the lesson on circumference of a circle.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Bernie and Tadarius have a goat at their grandfather’s house. The goat is allowed to walk around the yard twice a day. The goat is on a rope attached loosely to a tree. If the length of the rope from the tree to its longest point is fifteen feet, what is the circumference of the circle the goat could walk around the tree?
In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, what is the circumference of the circle that goat could walk around the tree?
Now that we have identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the circumference of the circle the goat would walk.
During this lesson we will learn about circumference of a circle to complete this SOLVE problem at the end of the lesson.
PARTS OF A CIRCLE
Cut out the circle.
What is the term we use to identify the distance around a figure? Perimeter
When we are working with a circle, we can use the term circumference to describe the distance around the circle.
Now fold the circle in half creating two congruent parts. The fold is the diameter. Trace along the diameter using a blue colored pencil and ruler and label it “diameter.”
Now, fold the circle along the diameter again and then fold to create four equal, or congruent, sections of the circle.
The point where all fold lines meet and/or intersect is the midpoint, or center, of a circle. Draw a small dot and label the midpoint using a green pencil.
The four line segments that extend from the midpoint to the circumference are each a radius.
Choose one radius, trace over it with an orange pencil, and label it “radius.”
RADIUS AND DIAMETER
Using the circle we marked with radius and the diameter, make a prediction about how many of the radii would equal the diameter of the circle. We can use a string to measure the radius of the circle and then the diameter.
Let’s measure!
What is the radius of the circle? About two point five inches
What is the diameter of the circle? About five inches
Describe the relationship between a radius and a diameter. A diameter is twice the length of the radius.
Take a look at Problem One on the next page.
Let’s measure the radius of the circle to the nearest inch with a ruler. Predict how long a diameter will be and measure to see if it is twice the length of the radius.
What information is given? The radius is six centimeters.
Without using the string or ruler, explain how you could determine the diameter. It is twice the radius.
What is the radius? Six centimeters
What is the diameter? Twelve centimeters
DIAMETER AND CIRCUMFERENCE
Lay the string across the diameter. Cut the string to show the length of the diameter.
What are some possible strategies for using the string to determine the length of the circumference? We could lay the diameter string, marking the length of the string on the circumference, then laying the string again.
How many diameter lengths did it take to go around the circumference of the circle? About three lengths of string.
Measure the string and then find the circumference in millimeters.
Lay the longer piece of string around the circumference of the circle on Problem One. Compare the length of the circumference to the length of the diameter.
Lay the string over the diameter, fold it one length, then lay again and fold. We should see that the length of the circumference is just larger than three times the diameter.
Find the length of the string to the nearest millimeter. Fill in the measurements for Problem One in the graphic organizer.
Complete the measurements of the remaining circles and fill in the organizer.
If we divide the circumference by the diameter, what do you estimate our answer will be? Explain your thinking. The quotient will be about three because we know from our activity that the distance around the circle was about three times the length of the string that measured the diameter.
Let’s find the quotient of the circumference and the diameter using a calculator.
CIRCLE CIRCUMFERENCE
Look at the values in the last column of the table.
What do you notice about all of the values? They are approximately three point one four.
This value of approximately three point one four is the value that will be the quotient every time we divide the circumference of a circle by its diameter.
The relationship value is identified as pi. This is the symbol for pi .
Pi is an approximate value because it is an irrational number. We usually round it off to three point one four when using it in a calculation.
Explain how we can write the relationship between the circumference and the diameter of any circle. Circumference divided by diameter is equal to pi.
How can we isolate the C (circumference) to rewrite the equation? We can multiply both sides by “d.”
What is the new formula we created to find the circumference of the circle? C (circumference) is equal to pi d or pi times the diameter.
If we are only given the diameter, is it possible to find the circumference? Yes, we can multiply the diameter times pi.
Complete rows two through four of the organizer. Remember that the radius should be multiplied by two before a value for “d” diameter can be substituted into the formula.
What is the radius of the circle? Two inches
Do we have enough information to determine the circumference of the circle? Explain your thinking. Yes, because we can find the value of diameter by doubling the radius. Then, we need to multiply the diameter by the value of pi. We know pi is a standard value of approximately three point one four.
FOLDABLE
We’re using the geometry foldable and we have folded all four corners into the center.
Write the information regarding the circumference of a circle on the back of the foldable.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson.
The question was, Bernie and Tadarius have a goat at their grandfather’s house. The goat is allowed to walk around the yard twice a day. The goat is on a rope attached loosely to a tree. If the length of the rope from the tree to its longest point is fifteen feet, what is the circumference of the circle the goat could walk around the tree?
S, Study the Problem. Underline the question and complete this statement. This problem is asking me to find the circumference of the circle the goat would walk.
O, Organize the Facts. First we identify the facts. Bernie and Tadarius have a goat at their grandfather’s house./ The goat is allowed to walk around the yard twice a day./ The goat is on a rope attached loosely to a tree./ If the length of the rope from the tree to its longest point is fifteen feet,/ what is the circumference of the circle the goat could walk around the tree?
Eliminate the unnecessary facts. We eliminate any facts that are not necessary to help us find the circumference.
Then we list the necessary facts. The rope is attached to a tree, and the length of the rope or the radius is fifteen feet.
L, Line Up a Plan. Write in words what your plan of action will be. Multiply the radius times two, which is the length of the rope, to get the diameter. Find the circumference of the circle by multiplying the diameter of the circle times pi or use the formula for finding circumference, which is Circumference equals pi d.
Choose an operation or operations. Multiplication
V, Verify Your Plan with Action. First estimate your answer. Our estimate is about ninety feet.
Carry our your plan. Use the formula for circumference and substitute in the values for pi, which is an approximate value of three point one four, and the value for the diameter,which is thirty. Our answer is approximately ninety-four point two feet.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because we are looking for the circumference of the circle the goat would walk.
Is your answer reasonable? Compare your answer to the estimate. Yes, because it is close to our estimate.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. The circumference of the circle the goat would walk is ninety-four point two feet.
CLOSURE
Now let’s go back and discuss the essential questions from the lesson.
Our first question was, explain the relationship between the radius and the diameter of a circle. The diameter is twice the length of the radius.
Number two, explain the relationship between the diameter and the circumference of a circle. The circumference is about three times the length of the diameter.
And number three, why is the circumference of a circle an approximate value? Justify your answer. The circumference is an approximate value, because the formula for the circumference is C equals pi d. Because pi is an irrational value, the circumference will not be an exact value.