THINK: Go back to the star-like image and manipulate it to get a very high r-ratio. What does the figure (image) look like? Try to get the ratio as high as you possibly can without crossing the lines. (Java File)

**All squares have r = 16. ****|
****Java
File**

**All circles have r = 4*pi = 12.56...
****| ****Java
File**

**All 4x5 (8x10) rectangles have r = 16. 2.
****| ****Java
File**

THINK: Triangles are figures that are 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? What about the ratio (r)?

After exploring general triangles we can explore "special" triangles:

**All 3-4-5 (6-8-10) rt. triangles have r = 24.
****| ****Java
File**

**All equilateral triangles have r = 12*sqrt(3) = 20.78
****| ****Java
File**

**All 45-45-90 rt triangles have r = 12 + 8sqrt(2) = 23.3.
****| ****Java
File**

**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?