TANGENTS ALONG THE DIAMETER
Along the diameter of a circle
you can construct circles with equal radii that are tangent to each other.
The outermost circles in the string of circles will be tangent to the large
circle. (Tangent means that the circles touch each other but do not cross
over each other, nor do they leave any gaps.)
How does the combined area of all of the shaded circles relate to the area
of the entire circle?
What is the relationship
between the area of small circles that are tangent to one another and the
end circles are tangent to one large circle? In addition, the extension asks what is the relationship of
the circumference of the larger circle to the sum of the circumferences of
the enclosed smaller circles?
Solve/Investigate the Problem
I am going to use Geometer’s
Sketchpad and let it do the “computation” for area and perimeter for
me. I will create several examples
using different numbers of tangent circles within a larger circle and see
if there is a pattern concerning the area.
Then I will measure the circumferences of the circles and compare to
see if there is a pattern or relationship.
of the Problem
I designed four different scenarios: three tangent circles along a diameter,
four circles, five circles and six circles. The results lead me to believe the following:
There is a pattern to determine the relationship of the
area between the combined number of tangent circles along a diameter of a
larger circle and the area of the larger circle. The area of the larger circle will be the same times larger
as there are circles. For example,
if you click
will notice that if a circle has three tangent circles, the area of the
larger circle will be three times the size of the combined areas of the
three smaller circles. Click here to see that the area of a circle with
four circles that are tangent along the diameter will be four times the
area of the combined four small circles.
Click here to see how it works for 5
smaller circles and click here to see 6
Extensions of the Problem
were to walk along the entire circumferences of all the small circles,
would you walk farther or less far than if you walked around the
circumference of the largest circle? How much more or how much less would
On the same Geometer’s Sketchpad pages, I calculated
the circumferences of the different examples. In all of the examples, the circumferences were equal to the
combined circumferences of the smaller circles.
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