Intermath | Workshop Support
Write-up
Title
Circle of circles
Problem Statement
Two larger circles with equal radii pass through each other's centers. A
smaller circle can be created inside the overlapping region so that it is
tangent to the other circles. (Tangent means that the circles touch each
other but do not cross over each other, nor do they leave any gaps.)
Compare the area and circumference of the smaller circle to the area and
circumference of the larger circle.
Problem setup
You need to construct two circles that have equal radii. You have to construct
the circles so that they will cross at their center point. You construct an inner
circle that will be tangent to the two larger circles. Remember that the two
circles cannot cross over each other. You need to find the area and the
circumference of the two larger circles. They should be the same. Now find the
area and the circumference of the smaller circle. Give a relationship between
the larger circle and smaller circle using their areas and circumferences.
Investigation/Exploration of the Problem
You need to find out how to create two circles. You have to know what the
tangent means and how to make the two circles cross their each other’s centers.
The smaller circle would seem to be 1/4th the area of the larger circle and 1/2
the circumference of the larger circle. You have to construct the circles and find
the correct measurements for each circle.
Plans to Solve/Investigate the Problem
You need to determine how to construct two circles with equal radii. You can
use geometer’s sketchpad to draw these circles. The first step would be to
construct a line segment. You would use this line segment to create the radius
of the first circle. You would then make the point and construct the first circle.
You would then put a point on the first circle and construct the next circle using
the center point and the point on the circle. You would then need to draw a line
segment between the points that the two circles would cross. You find the
midpoint and then construct the smaller circle. You go to the measurement and
find the circumference and the area of all the circles. The area of the larger
circle is four times the area of the smaller circle. This would go along with the
formula for finding the circumference of the circle. The circumference of a
circle can be found using the formulas C = 2*pi*r or C = pi*d (* means to
multiply, r = radius, and d = diameter). The area of a circle can be found using
the formula Area= Pi x r2
where A is the area and r is the circle's radius. If you use these two formulas
then the circumference of the smaller circle should be 1/2 the circumference of
the larger circle. The measurements show that this is a true statement.
Circumference c1 = 16.96 cm
Circumference
AB = 8.48 cm
Area c1 = 22.88 cm2
Circumference
AB = 8.48 cm
Area c1 = 22.88 cm2
Area
AB = 5.72 cm2
B
A
c1
Investigation/Exploration of the Problem
Carry out your plans/strategies you planned initially. Give a well organized
explanation and details about how the problem was approached and explored.
You should do that so that the reader can follow/construct/understand your work
with minimal effort. Include numerical, graphical data (to the extend that it is
applicable) to support your arguments and conjectures. Try to include multiple
approaches/representations (numerical, graphical and symbolic) to the problem
and the solution. Label diagrams, tables, graphs, or other visual representations
you used. Provide an algebraic proof/solution for your conjectures/observations
where it's applicable.
Extensions of the Problem
Discuss possible extensions for the problem and explore/investigate at least one
of the extensions you discussed.
Author & Contact
Carole C. Jackson
carolecjackson@yahoo.com
cjackson@yahoo.com
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2
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