Circle Area Formula
Circle Area Formula
Teacher Notes
Introduction
The primary aim of this activity is to support the conceptual understanding of where the formula
of Area = πr2 comes from. Instead of presenting the formula as is, students can explore the
limiting process of slicing a circle up into progresively more and more sectors.
Physically cutting up a circle with scissors into sectors and then rearranging these pieces has
long been a traditional approach in the classroom, but often students only saw one example, and
then had to generalise from that. This activity gives them access to all the other examples
required to better appreciate and understand the extension of this idea.
Whilst it is easier–– and often only necessary–– for students to know how to process the formula
Area = πr2, this activity applies knowledge of areas of rectangles, parallelograms and trapeziums
to working out the area of a new shape, the circle.
The idea of a limit tending towards an exact answer will be required later in their mathematical
career, primarily in the topics of geometric series and calculus. This activity serves as an early
introduction to this important mathematical technique.
Resources
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Single tns file, called “CircleAreaFormulav2.tns”
Plain or squared paper.
Ruler and pair of compasses, if desired.
TI-Nspire skills required
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Move from one page to the next
Click on minimised sliders to increase/decrease a number’s value.
Grab and drag a single point
The activity – suggestions for class use
Problem 1
Page 1.1 outlines the activity
and lists the contents of each
page.
© 2010 Texas Instruments Education Technology
CircleAreaFormulav2
Circle Area Formula
Problem 2
Page 2.1 outlines the purpose of pages 2.2 to 2.5.
Students can check their answers to each page by pressing
/ then `©
The screenshot below shows the output for a wrong
response.
Students should not progress
until they have all four
responses correct!
Problem 3
Page 3.1 introduces the main idea behind this activity.
It initially requires students to apply their knowledge of
working out areas of trapezia and parallelograms, depending
on whether there are an odd or even number of sectors.
The slider for sectors has
been restricted to values
between 5 and 9 to ensure
that around the classroom,
several students are tackling
the same problem.
It is possible to edit the
number of sectors directly by
double clicking on the
number itself, and changing it
to any (large) integer value.
When verifying the rough area calculations, students will most likely employ a wide variety of
techniques.
Notes:
© 2010 Texas Instruments Education Technology
CircleAreaFormulav2
Circle Area Formula
1. The radius of the circle can be altered by grabbing the
circumference of the circle when it is highlighted in bold,
as shown on the right.
2. If they wish, students can insert a Calculator Page 3.3
into the document by pressing / ~ then 1.
3. The displayed ‘height’ of the rearranged sectors
(here 6·6 u) is not the radius of the circle. It is measured
between the extremes of each arc. However, as the
number of sectors increases, this length converges to the
circle’s actual radius.
4. The displayed length of the rearranged sectors
(here 20.2 u) is measured between the extremes of the
sector corners. It is not a simple fraction of the circle’s
circumference. However, as the number of sectors
increases, this length converges to half of the circle’s
circumference.
If students are stuck, suggest that they consider the base
length of the trapezium to be made up of 3 ‘bits’, as
highlighted on the left.
So, each ‘bit’ is length 19·2 ÷ 3= 6·63
Therefore the top length of the trapezium is 6·63 × 2 = 13·26
Area of trapezium = ½ × (19·9 + 13·26) × 6·7 = 111·11 unit2
Similarly, a hint for the parallelogram situation is for the
students to consider each sector being split vertically, as
shown on the left.
The length of 19·7 is longer than the required base length of
the parallelogram. We only need about 76 of 19·7 = 16·89.
Area of parallelogram = 16·89 × 6·4 = 108·07 unit2
Problem 4
Problem 4 extends the geometric principles from problem 3 to
situations when the number of sectors have increased.
Essentially, the students need to notice that the trapezia and
parallelograms become closer and closer to a rectangle.
© 2010 Texas Instruments Education Technology
CircleAreaFormulav2
Circle Area Formula
The radius of the circle can
be altered by grabbing the
circumference of the circle
when it is highlighted in bold,
as shown on the right.
Problem 5
Problem 5 examines the numbers behind the process that
was met in problem 4.
Essentially, the students need to notice that the rough area of
the rearranged sectors converges on the true value of the
circle’s area, as the number of sectors increases.
In this problem, the radius of the circle cannot be changed.
Problem 6
Problem 6 draws together everything met so far, in readiness
for deriving the circle area formula itself.
Page 6.2 contains 4 statements, and the task is to tick only
those statements that are true.
Students can check their
understanding by pressing
b2
They should not carry on until
they have answered this
page correctly!
© 2010 Texas Instruments Education Technology
CircleAreaFormulav2
Circle Area Formula
By page 6.3, students should be able to apply their recently
developed knowledge to the ‘limit’ situation of a circle split
into an infinite number of sectors.
Authors
Nevil Hopley, Head of Mathematics, George Watson’s College, Edinburgh.
Barrie Galpin, Author & Editor.
January 2010.
© 2010 Texas Instruments Education Technology
CircleAreaFormulav2