LESSON #6 Circles

Last revision date - July 25, 2012
Revision date - Fall 2017

Purpose: This lesson introduces the relationship between a polygon and a circle to draw “circles” and “arcs” in the Logo workspace.
Essential Question: How can Logo be used to draw “circles” and arcs of given dimensions (circumference or diameter). What is the difference between arc measure and arc length?

A Section of the Lesson Follows

In the Logo programming language, the turtle can move in two directions and can turn right or left. Is it possible for the turtle to draw what appears to be a circle? You may have noticed that a turtle step (pixel) of 1 unit is very small and can be thought of as a single point.
Pretend you are the turtle and stand in a large room, how would you “walk” a circle? Go ahead and try this! You probably took one step, turned a very small amount, took another step, and turned a very small amount (in the same direction) until you reached your starting point. Since the turtle steps are so small, a polygon with over 100 sides will appear to look like a circle. For discussion purposes we will call this shape a regular polygon or a Logo “circle” instead of a 100-gon. The first two examples will draw “circles” and the third example will draw a semi-circle.
Example 1
The larger circle in Figure 6.1 was drawn using repeat 360 [fd 1 rt 1].
This repeat command will result in a 360-sided polygon, because the turtle turned 1 degree 360 times.
An examination of the repeat command tells me that the circumference is (360 * 1) or 360 turtle steps
The circumference formula is C = Pi * d, so the diameter of the larger circle is found by solving for d.
d = 360 / Pi = 114.6 steps
Example 2

The smaller circle in figure 6.1 was drawn using the repeat 180 [fd 1 rt 2] command.
This repeat command will result in a 180-sided polygon, because the turtle turned 2 degrees 180 times. The Circumference of the smaller circle is C = Pi * d.
d = 180 / Pi = 57.3 turtle steps
The astute reader will notice that the diameters for the two circles have a ratio of 2 to 1. A common problem found in school mathematics is to find the ratio of the area of the two circles. While these lessons have not included problems on area, this is a great opportunity to look at the ratio of similar circles. Will the ratio of the areas also be 2:1?

Omitted section of this lesson

The real challenge is when you are given the dimensions first and need to write the Logo code. For example what if you want to code a 90-degree arc for a circle that has a radius of 70 steps?
First, I am going to determine the amount of turn. The easiest way would be to set up the repeat template as follows:
Repeat 90 [fd ? rt 1]

The next step is to determine the circumference of the full circle. Since the radius is 70 steps the diameter is 140 steps. The circumference would be (140*Pi) or 439.6 steps.
This problem requires a quarter circle. This means the Circumference needs to be divided by 4 to determine the number of turtle steps (arc length). 439.6 / 4 = 109.9 steps
The last step is to calculate the missing value in the forward command: 90*? = 109.9 so ? = 1.22 steps
The result will be Repeat 90 [fd 1.22 rt 1]