1. ACE problems
Extension 17

2. Problem
Suppose you tie together the ends of a piece of string to form a loop that is 30 units long.
A unit is equal to the
length of a side of one
floor tile.

3. Wave 1
Suppose you arranged the string to form an equilateral triangle. What would the area of the enclosed space be?
What would the area be if you formed a square?
What would the area be if you formed a regular hexagon?

4. Wave 2
Of all the rectangles with a perimeter of 30 units, which has greatest area?

5. Wave 3
Of all the triangles with a perimeter of 30 units, which has greatest area?

6. Wave 4
How does the area of a regular octagon with a perimeter of 30 units compare to the areas of a triangle, square, and a hexagon with perimeters of 30 units?

7. Wave 5
What happens to the enclosed area as the 30-unit perimeter is used to make regular polygons of more and more sides?

8. Wave 6
As the number of sides of a polygon gets larger and larger, what shape does the polygon eventually resemble?