Playful Geometry: Turning Circles Into Squares

Have you tried the Playful Geometry activity, Turning Circles Into Triangles?  Would you ever guess that a smooth, curved shape could give rise to one of angles and straight lines?  It’s pretty neat, right?  Today we’ll expand on that activity and play around with turning circles in to squares.

Are you ready to give it a try?  We’ll look at two different ways of creating squares from circles.  In the first example, the circles and points are provided for you on our printable.  This would be the most appropriate method to use with younger children (ages 6-9 years).  The second method works well for older children (10+ years) who are practicing or are comfortable with their compass  skills.

Before starting, gather your supplies:

• a copy of our printables or
• graph paper
  
• a ruler
• a pencil
• a compass with pencil

If you’d like, you can review the characteristics of a square with your child:

• All four sides are equal in length
• The opposite sides are parallel to each other
• All angles are equal
• If you were to divide a square in half diagonally, the dividing line would bisect its angles.

Method 1
Before you begin following the directions on the printable, notice with your child the arrangement of circles and dots.  How many circles are there?  Are they all the same size?  How are they arranged (touching, overlapping)?  How many points are there?  Where do the points lay?  Follow the directions to connect the points and create a square.  Why did that work?

Now move on to the second set of circles.  How many circles are there now?  Once you have created squares following the directions, you can take it a step further by connecting the blackpoints with diagonal lines.  Did that create any more squares?  How many total squares can you make with this set of five circles?

Method 2
1. Create a circle on a piece of graph paper by placing the point of the compass on a place where two lines intersect.  When the circle is complete, darken the center point with your pencil.
2. Now place the point of the compass on the circumference of the circle horizontally from the center point of the circle.  You can follow the line on the graph paper.  Draw a second circle and again darken the center point.  Each circle should cross through the center of the other.
3. Using a ruler, connect the two center dots with a horizontal line.  Now draw a vertical line to connect where the two circles intersect.
4. Place the point of the compass where the horizontal and vertical lines intersect.  Place the pencil of the compass on the center point of one of the circles and draw another circle.  With a ruler, connect the four points where this third circle crosses the horizontal and vertical lines.  It should look like this…

You can also follow the instructions in this Khan Academy video.

If you wanted to create even more squares, where would you add more circles?  Can you come up with other arrangements of circles that would allow you to draw squares?  Go ahead…  Have fun playing around with it and see what you can create.