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### Squares and Circles: How Many Intersections?

```
Date: 02/27/2002 at 18:52:25
From: Ashley S.
Subject: Squares and Circles

Our algebra class was given a problem on a test asking us the greatest
possible number of times a square's edge can intersect with a circle
when it is placed on top. They can be any size.

can be done in 8 different places, if done at the perfect sizes. We
were wondering if you can solve this disagreement.

Thank you,
Ashley S. and 4th hour algebra
```

```
Date: 02/27/2002 at 20:10:17
From: Doctor Ian
Subject: Re: Squares and Circles

Hi Ashley,

Try this:

1. Draw a circle.

.
.         .
.             .

.               .

.             .
.         .
.

2. Draw a square whose center is also at the center of the circle,
and which is just big enough so that the circle touches it at
the center of each of its sides. (This is the smallest square
that will fit completely around the circle.)

+-------.-------+
|  .        .   |
|.             .|
|               |
.               .
|               |
|.             .|
|  .         .  |
+-------.-------+

3. Draw another square whose center is the center of the circle,
and which is just big enough to touch the circle at its
corners.  (This is the largest square that will fit completely
inside the circle.)

+-------.-------+
|  .        .   |
|.+-----------+.|
| |           | |
. |           | .
| |           | |
|.+-----------+.|
|  .         .  |
+-------.-------+

4. Now draw _any_ square whose center is at the center of the
circle, smaller than the first square, and larger than the
second:

+-------------+
|.............|
|. +-------+ .|
|. |       | .|
|. |       | .|
|. +-------+ .|
|.............|
+-------------+

In how many places does this square intersect the circle?

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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