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# Polyarcs

By Henri Picciotto

Consider the two pieces that are created by drawing a quarter circle with radius one, centered at a corner of a unit square, and cutting along the resulting arc:

We will call these pieces monarcs.

- What is the perimeter and area of each monarc?
- What is the average area of the two monarcs?

Two monarcs can be put together to create diarcs in seven distinct ways. Here they are:

- What is the perimeter and area of each diarc?
- What is the average area of the diarcs?

All the monarcs and diarcs can be combined into a 2 by 4 rectangle, or a pleasingly symmetric curvilinear figure.

- What is the area and perimeter of each figure?
- Find all the triarcs.
- Find interesting figures made of diarcs and triarcs.

## Discussion

- When does the area of a polyarc figure involve pi? When does it not?
- When does the perimeter of a polyarc figure involve pi? When does it not?
- In #5, some students argue that since both figures consist of the same pieces, they must have the same perimeter and area. Explain why they are right or wrong.