EZ Math Movie, Tutorials, Turtle Graphics, Drawing a Square

The turtle walks the square.

Think about walking the turtle along the perimeter of a square and watch this animation:

The turtle draws a square animation

A square has 4 sides, 4 corners.

Note that the turtle does the same thing 4 times: It goes down a side and turns a corner. The code could look like this:

forward(5); // first side
left(90);   // first corner

forward(5); // second side
left(90);   // second corner

forward(5); // third side
left(90);   // third corner

forward(5); // fourth side
left(90);   // fourth corner

Using a loop

Since we are going to do the same thing 4 times, we will use a loop that cycles 4 times to draw the square. Every time through a loop cycle we will draw one side and turn one corner. Here is how the code looks when a for loop is used, note the loop counter variable is named side:

// This loop runs 4 times.
for(side = 0; side < 4; side++)
  // Go down a side.

  // Turn the corner.

The last left(90)

Perhaps the last command of left(90) may not seem necessary. If it were somehow skipped, the square would still look the same on the graph. So why turn the last corner when the drawing seems already done? A very good reason for this is that it leaves the turtle in the same state that it was in at the start of the drawing. It started out at a certain location aimed right, and it finishes the drawing at the same location aimed the same way, right. When drawing closed figures like the square, this way of doing things makes it easy to link simple drawings together in order to create complex drawings. So it is often handy to turn one last time in order to get the turtle back to how it was at the start.

Not the turning angle

Be sure to understand that the 90 degree turns that the turtle is doing are not the 90 degree internal angles to the square.

Not the turtle's turning angle

The actual turning angle

The turtle's 90 turning angle in the left(90) command is actually an external angle to the square:

The turtle's turning angle

360 degree total turn

If we forget about the turtle's movement along the sides of the square, and just look at its turnings, we will notice that it turns through a total angle of 360 degrees as it goes all the way around the square. Four left(90) commands add up to a total spin of 360 degrees:

The turtle's total turning angle

4 sided regular polygon

The square is simply a 4 sided regular polygon. Let's see how we could calculate the turtle's turning angle for a regular polygon.

(turning angle) = (360 degrees) / (number of sides)

For a square that would be:

(90 degrees) = (360 degrees) / (4)

In general, for a regular polygon the turtle's turning angle will equal the total turning angle for the polygon, 360 degrees, divided by the number of sides. In the case of the square, the turtle will turn a total of 360 degrees as it moves around the square, and at every corner, or vertex, it will turn 1/4 the total turning angle. That is, 1/4 of 360 degrees equals 90 degrees.

We will use this thinking in the next tutorial.

The example program

Angle measurement: Degrees Radians

Go on input

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Next tutorial: Drawing a polygon


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All contents of EZ Math Movie are copyrighted: Copyright 2011, Edward A. Zobel. All rights reserved.

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