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Title
Relating Areas

Problem Statement
Draw a circle with radius = 2 units. Draw a square (BEST) around the circle so that each side of the square touches the circle. Draw perpendicular diameters so that the circle and the square are each cut into four equal parts. The circle is cut into four equal sectors while the square is cut into four equal squares. See actual intermath investigation.

What is the area of square BEST? Of square COAT?

Let’s call the radius of the circle r. In terms of r, what is the area of square BEST? Of square COAT? Of square BDOC? Of Rectangle CTSM? Of the concave hexagon CODEST?

Is the area of the circle smaller of larger than the area of square BEST? Why? Is the area of the circle smaller of larger than the area of concave hexagon CODEST? Why?

How can we use this situation to show that the area of a circle equals approximately 3r^2?

Problem setup
How can we come to a better understanding about what the area of a circle is, that is the radius and the pi, by comparing it to what we know about the area of a simpler object, such as a square. Draw a square with a circle inscribed inside so that each side of the square touches the circle. Draw the radius of the inner circle and label it r. What is the area of the outer square in terms of r? How does the area of the square in terms of r compare to the area of the circle? What is the area of the square not covered by the circle? How does this area compare to the area of the circle? To the area of the square? How are the areas of the circle and the square related to each other in terms of r? How can we use this situation to show that the area of a circle is equal to approximately 3r^2? This problem is similar to others that I have solved in that I come to a better understanding about the properties of a shape through manipulations and links to prior knowledge. It is also similar to other problems I have solved earlier this semester with my class mates about the area of a circle.

Plans to Solve/Investigate the Problem
The first thing I plan to do is visualize a circle inscribed in a square, what it would look like as well as how their areas would compare to each other After I think I know what the figure would look like, I will begin drawing and labeling it according to the investigation instructions. Once I have the figure drawn and labeled, I will calculate the areas of the circle and square in terms of the units given by the perpendicular lines as well as in terms of r. After I have calculated the areas of the circle and square in terms of r, I will make conjectures about the relationship between the two areas in terms of r. Finally, I will explain how to use this situation to show that the area of a circle is equal to approximately 3r^2?

Investigation/Exploration of the Problem
To solve this problem, I started by visualizing a circle inscribed in a square, thinking about what it would look like as well as how their areas would compare to each other. After visualizing, I began to draw and label the figure according to the investigation instructions. Once every thing had been drawn and labeled, I calculated the areas of the squares BEST and COAT, as well as the circle OA using the units of the two perpendicular lines. Square BEST was 16 square units, square COAT was 4 square units and circle OA was equal to approximately 12. 6 square units. The area of square BEST was greater than the area of circle OA. After I had made my calculations, I analyzed the figure to see if I noticed any thing. Upon analyzing the figure, I noticed that the radius of the circle was equal to half the base as well as half of the height of the square. This was a very important observation because I could now use this information to find the area of square BEST in terms of r. Since r was equal to half the base of BEST, the base of the BEST was equal to 2r. In the same way, the height of BEST was also equal to 2r. Once that I knew the dimensions of square BEST in terms of r, I was able to calculate the area of BEST in terms of r. The area of square BEST is equal to its base times its height. The area of square BEST, in terms of r, is equal to 2r times 2r or 4r^2. In the same way I also found the areas of squares COAT and BDOC, rectangle CTSM, concave hexagon CODEST, and the circle OA in terms of r. The squares COAT and BDOC were both equal to r^2. The rectangle CTSM was equal to 2r^2. And the concave hexagon was equal to 3r^2. The most exciting discovery I made was when I calculated the circle in terms of r. We all know that a circle is equal to pi*r^2, but what does that mean? Instead of writing pi in the equation for the area of the circle, I decided to substitute pi with its actual numerical quantity 3.14… To my surprise the area of the circle was only slightly greater than the area of the concave hexagon CODEST. Upon analyzing the figure to see why, I realized that the area of CODEST was 3r^2 and the area of the circle OA was 3.1r^2, which is only about a half of a square cm difference. Knowing this information I could then generalize that the area of any square is approximately equal to three times its radius squared.

Extensions of the Problem
Visualize a sphere inscribed inside of a cube so that each side of the cube touches the sphere. How does the volume of the sphere compare to the volume of the cube?

Author & Contact
Nathan Little.
nlittle@aug.edu
My comments to Dr. Crawford
Link(s) to resources, references, lesson plans, and/or other materials
My constructions
Intermath

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