THINK: We have presented the ratio of r = p^2/a as an indicator of cell structure. Visualize an image of what this cell might look like. What does it look like?
THINK: Why does the calculation r = (p^2)/a include the perimeter squared and not the perimeter?
THINK: Can you make this figure look VERY Bizarre? (Java File) How high can you make the ratio, r without crossing the lines? Try to make it as high as possible.
THINK: Let's take the ratio of a standard square of side 4. What is the ratio, r = (p^2)/a? Can you prove that every square has a ratio, r of 16? [Hint]
THINK: What is the ratio of a circle of radius 3 units? How about with radius 7 units? What about radius 2.34 units? Prove that the ratio, r of a circle is 12.56. [Hint]
THINK: What is the ratio of a rectangle with length l and width w? [Hint]
THINK: What is the ratio, r for a rectangle with width 4 and length 5? What about a rectangle with width 8 and length 10? What about any multiple of such sides? Prove that a rectangle of width 4 and length 5 has a ratio, r of 16.2. [Hint]
THINK: Triangles are figures unique in this ratio and are relatively easy to explore. The sketch on the next page offers the ability to explore a triangle between two parallel lines. Carefully note the area of the triangle as you move the top vertex, what happens to the perimeter? Why does the ratio, r increase but the area remains the same? Why does the area remain the same as the perimeter increases? [Hint]
Conclusion: We have explored the cancer ratio (r) with figures up to four sides. Establish the ratio for sides with more than four and try to create a figure with a large ratio and small ratio. What types of figures produce the largest ratio? What types of figures produce the smallest ratios?
Think: Try various ratios out with this figure to experiment with the topics you just learned?