Title
Circle in a Semicircle
Problem Statement
A circle is inscribed inside of a semicircle.
How do their areas relate. How
does the arc length of the semicircle relate to the circumference of the smaller
circle?
Problem setup
What is the relationship
between a semicircle and a circle inscribed within? What do you notice about their areas? How does the circumference of the smaller
circle compare to the arc length of the semicircle? This problem is similar to others I have done
in that I learn about the properties of a circle through manipulations. It is also similar to a problem I have done
dealing with circumference and area of a circle.
Plans to
Solve/Investigate the Problem
To solve this problem I will first visualize a circle inscribed inside
a semicircle and what it would look like.
Because of the dimensions of these circles I will construct the figure
on Geometer’s Sketchpad and calculate their areas, arc length, and
circumference using the computer. I will
then drag on the figure to make it larger and smaller, paying attention to the
resulting area, arc length, and circumference.
Finally, I will try to make some conjectures about the relationship
between the area of the circle and semicircle as well as the relationship
between the circumference of the small circle and the arc length of the
semicircle.
Investigation/Exploration
of the Problem
I will begin by visualizing a circle inscribed within a
semicircle. I will then construct the
figure and calculate areas, the arc length of the semicircle, and the
circumference of the small circle using Sketchpad. After constructing the figures and making my
calculations I have concluded that the small circle has exactly half the area
of the semicircle it is inscribed within.
I have also concluded that the circumference of the small circle is
congruent to the arc length of the semicircle.
Extensions of the Problem
How would the figures be related if the problem involved a sphere
inscribed inside of a hemisphere?
Author & Contact
Nathan Little
nlittle@aug.edu
Link(s) to resources, references,
lesson plans, and/or other materials
My constructions
Intermath
U Important Note: You should compose your write-up targeting an audience in mind rather than just the instructor for the course. You are creating a page to publish it on the web.