The number is defined to be the ratio circumference/diameter for any circle. We also all know that the area of a circle is 
. It is a sheer coincidence that they are both the same 
, even though one concerns the circumference and one concerns the area? No!
What about for a regular pentagon? It turns out that for the appropriate definition of the "diameter" of a regular pentagon, if we define the number θ (read as tetha) to be the ratio of the perimeter/diameter of a regular pentagon, then its area is always 
, where r is half of the diameter. For this to be true, what should be the "diameter" of a regular pentagon?
A. The distance between the farthest corners of the pentagon.
B. The diameter of the largest circle the fits inside the pentagon.
C. The diameter of the smallest circle that fits around the pentagon.
D. The distance from the base to the opposite corner of the pentagon.
E. Other, not easy to describe.
F. It's a trick question.
Choose an answer from A through F and explain your reasoning in detail.
The number is defined to be the ratio circumference/diameter for any circle. We also all know that the area of a circle is . It is a sheer coincidence that they are both the same , even though one concerns the circumference and one concerns the area? No!
What about for a regular pentagon? It turns out that for the appropriate definition of the "diameter" of a regular pentagon, if we define the number θ (read as tetha) to be the ratio of the perimeter/diameter of a regular pentagon, then its area is always , where r is half of the diameter. For this to be true, what should be the "diameter" of a regular pentagon?
A. The distance between the farthest corners of the pentagon.
B. The diameter of the largest circle the fits inside the pentagon.
C. The diameter of the smallest circle that fits around the pentagon.
D. The distance from the base to the opposite corner of the pentagon.
E. Other, not easy to describe.
F. It's a trick question.
Choose an answer from A through F and explain your reasoning in detail.
