Area of Circles
This lesson is based from the textbook Holt Middle School Math: Course 2.
J.M., Chard, D.J., Jackson, A., Milgram, J., Scheer, J.K., Waits,
B.K. (2004). Holt middle
school math: Course 2. Orlando: A Harcourt
Lesson Title: Finding Areas
Grade Level: 7th grade
Course Title: Compacted Math
Time Allotted: 1
Number of Students: 24-34 students
Information About Students: None
According to the NCTM Principles and Standards of
following standards are met in this lesson:
1. To understand measurable attributes of objects and
the units, systems, and processes of measurement.
- Understand both
metric and customary systems of measurement.
relationships among units and convert from one unit to another within
the same system.
select, and use units of appropriate size and type to measure angles,
perimeter, area, surface area, and volume.
To apply appropriate techniques, tools, and formulas to determine
and use formulas to determine the circumference of circles and the area
of triangles, parallelograms, trapezoids, and circles and develop
strategies to find the area of more-complex shapes.
According to the NCSCOS, the following
standards are met in this lesson:
2: The learner will demonstrate an understanding and use of the
properties and relationships in geometry, and standard units of metric
and customary measurement.
- For the
students to understand through construction how to find the area of a
- For the
students to see the relationships between the area and circumference of
and/or Use of Space:
Math Fact of the Day:
- Cut out
circles for each group with diameter of about 12 inches
do you call a crushed angle?
- Share this
with the students at the beginning of class. It is similar to
when an English teacher shares a quote at the beginning of language
Problem of the Day:
16 in. pizza sells for $11.99. A 10 in. pizza sells for
$5.99. Which size gives you more pizza per penny? Explain.
- Give this
to the students at the beginning of class and have them turn in their
answers with their unit portfolio at the end of the unit. The
answer for this problem is: The
larger pizza because you get more than twice as much (64pi square
inches instead of 25 pi square inches) for about twice the price.
problem is an introduction into the day's lesson.
students should be able to solve this problem after today's lesson.
will learn about how to find the area of circles today.
*NOTE: Throughout this lesson, I use pi to symbolize 3.14 and the
^ sign is used to show that something is raise to a power. For
example, r^2 means "r squared".
- To begin
with, ask the students to describe what they would need to know to plan
a pizza party for the class.
- Lead the
students into a discussion about how the size of the pizzas ordered
affects the number of pizzas needed to feed the class. Discuss
that you can find the best deal by comparing the unit costs per square
inch of two differently sized pizzas if you can determine the area of
each pizza, a circular figure.
- This is
where our lesson for today comes in handy because we will be discussing
how to find the area of a circle.
Yesterday we learned how to find the area of triangles and
trapezoids. Today we are going to discuss how to find the area of
circles. We are going to do this first by deriving the formula
for the area of a circle ourselves.
This activity was found at http://www.education-world.com/a_tsl/archives/00-2/area_of_circle_doc.shtml.
The objective of this activity is for students to form a rectangle by
partitioning a circle and relate A = bh to A = pi r^2.
break students into groups of 3. Do this by their last names
today - all of the last names that begin with A are in a group, all of
the last names that begin with B are in a group, and so on. Make
sure that there are no more than 3 students in each group.
with students practical applications for finding the area of a circle.
Explain the problems associated with partitioning a circle into unit
squares to find its area. (It's not exact). Give suggestions on
how the area might be determined.
- Pass out
the paper circles, scissors, rulers and markers/crayons.
- Have students
draw a diameter (it does not need to be exact), and use two different
colors to fill in the resulting semicircles.
students to cut the circle in half along the diameter. Then have them
cut each of the resulting semicircles into four equal sectors (They
should know what a sector is, but remind them that a sector is a region enclosed by two
radii and the arc joining their endpoints). There are now a total of
eight pieces, four of each color.
- Ask students to
assemble the eight pieces so that they form a shape which resembles a
rectangle. Provide them with the hint that the same colors should not
touch. (The resulting shape consists of sectors "pointing" in opposite
directions, side by side).
- Ask for
suggestions as to how to make the shape more like a rectangle. (This
can be achieved by cutting each of the sectors in half, again).
Give them hints to try to arrive at this answer.
- Have students
cut each of the sectors in half, once more, resulting in a total of 16
equal sectors, eight of each color. Solicit suggestions as to how to
make the shape even more like a rectangle. (This can be achieved by
cutting each of the sectors in half over and over again). Note: Do not
allow students to create more than 16 sectors since they can become
unmanageable. This would get too out of control.
- Ask students to
again assemble the sectors "pointing" in opposite directions, side by
side. Make sure that none of the same colors are touching.
- Ask students to
equate the parts of the approximated rectangle to the parts of the
original circle. The remainder of the lesson involves the mathematical
derivation of the formula for the area of a circle.
- The base, b, of
the rectangle is equivalent to half of the circumference, C. The
height, h, of the rectangle is equivalent to the radius, r, of the
circle. Therefore, using the formula for the area of a rectangle, A =
bh, we get b = C/2 and h = r.
- So the formula
for the area of the circle is now A = C/2*r.
- However, we know
that the circumference of a circle is equal to the diameter multiplied
by pi (d*pi). Thus, the formula can now be written as A =
- Since the
diameter is the same thing as twice the radius (2r), the formula can
now be written as A = 2r*pi /2*r.
- Simplifying this
equation, we arrive at A = r*pi*r or A = pi*r^2 as the area of a
- Great job!
We figured out the area of the circle ourselves. Now let's look
more into this formula and do a few examples together.
Here is a chart similar to those we looked at the past two days to help
you understand how to find the area of a circle, along with a picture
to help you learn.
AREA OF A CIRCLE
The area A of a circle
is the product of
pi and the square
of the circle's radius r.
*Make sure that the students are reminded that the variable r in this
formula works as a placeholder for the actual value of the
radius. Since r is squared, you must also square the value.
We cannot forget to square our value r.
Example 1: Find
the area of the circle.
*NOTE: Remind the students that pi is approximatley 3.14, thus,
we will substitute 3.14 for pi when calculating the area for circles.
We use the formula and substitute for r.
A=(3.14)(9 square meters)
A= 28.26 square meters.
Thus, the area of the circle is 28.26 square meters..
- Note: Our units are in square units. Make sure to
point this out to students and to explain that because we do not have a
specific unit of measurement to use, that we assume the measurement to
be in units.
Example 2: Find
the area of the circle.
*NOTE: Be careful - we are given the diameter, not the radius.
Since the radius is 1/2 of the diameter, we can find the radius by
So, r=10/2=5 cm
Now let's find the area of the circle.
A=78.5 square centimeters.
Thus, the area of our circle is 78.5 square centimeters.
Example 3: There
are circular fields located in the Wadi Rum Desert in Jordan. On
these fields, crops are grown on circular patches of irrigated
land. If the radius of one irrigated field is 60 feet, what is
the area of the field? Round your answer to the nearest whole
So, we know that r=60 feet. Let's find the area!
A=11,304 square feet.
Thus, the area of the field is is about 11,304 square feet.
- Journal: Students are
asked to complete this journal entry for the day: How do you
think perimter and circumference are important in the real world?
How about finding the areas of parallelograms, areas of triangles and
trapezoids, and areas or circles? Can you think of any jobs where
these mathematical concepts may be applied? Please give at least
2 examples of where you could apply the new mathematical concepts that
you have learned to the real world. Be thorough in your examples
and explain the applications of these concepts to the real world.
an extension activity is needed, the students can think and write
responses to the following questions:
finding the area of a circle when given the radius with finding the
area when given the diameter. What is the difference?
how to find the area of a circle with a diameter of 3 feet.
- Give an
example of a circular object in this classroom. Tell how you
could estimate the area of the object, and then estimate the are of the
object. You may use a ruler if needed.
you can see, we have learned how to find the area of circles. We
first derived this formula by starting with a circle and manipulating
the circle to see how we could find the area of the circle. We
discussed at the beginning of class the example with the pizza and how
we could find the better buy simply by finding the area of each pizza
and comparing the prices. This is something that you would use in
your everyday lives. Tomorrow we are going to discuss how we can
apply the concepts that we have learned in the past week to the real
will complete the problem of the day. Their completion
of this problem will show the teacher whether or not they understand
what they have learned in class that day. This activity also
teaches the students more
about mathematical communication.
students will also have a worksheet to complete. If the worksheet
is not completed in class, it is to be finished as homework. This
worksheet will count as a homework grade.
students are expected to participate in the activity, too. This
will be part of their classwork grade and their behavior/conduct grade.
are also expected to complete the journal entry for today, too.
This is part of their journal/problem of the day grade.
- What would you
- What would you
do the same?
- Was this a good
lesson - why or why not?