Triangles and 3D Shapes in Turtle Art

1Inside angle or turning angle?

The angle inside the corner of an equilateral triangle is 60°. But the turtle does not turn by the inside angle. It is walking straight, and at the corner it must turn by the angle outside the shape:

turning angle = 180 − inside angle
equilateral triangle: 180 − 60 = 120

Check: 3 turns × 120° = 360°, a full circle. So the correct program is:

repeat 3
   forward 100
   right 120

2A right-angled triangle

Let us draw a right-angled triangle with two equal sides. Its inside angles are 90°, 45° and 45°. The turns are:

• at the 90° corner: 180 − 90 = 90
• at each 45° corner: 180 − 45 = 135

Check: 90 + 135 + 135 = 360. ✔

forward 100
right 90
forward 100
right 135
forward 141
right 135

Why 141? The slanting side (the one opposite the right angle) is always the longest. On paper, draw two lines of 10 cm at a right angle with your ruler and protractor, then measure the slanting side: it is about 14.1 cm. So in steps, 100 and 100 give a slanting side of about 141.

3Making shapes look 3D

A screen is flat, but we can trick the eye into seeing depth. The trick: draw the front face, draw the back face the same size but moved up and to the right, then join the matching corners with slanting lines.

A cube (all faces are squares). Start at the bottom-left corner, pen down, facing up:

1. Front square: repeat 4 [forward 100, right 90].
2. Use set xy to move to (40, 40). With the pen down, this draws the first slanting line.
3. Back square: repeat 4 [forward 100, right 90] again.
4. Draw the other three slanting lines: pen up, go to a front corner, pen down, set xy to the matching back corner. The pairs are (0, 100) to (40, 140), (100, 100) to (140, 140) and (100, 0) to (140, 40).

A cuboid is the same, but the faces are rectangles: repeat 2 [forward 60, right 90, forward 120, right 90].

A cylinder has an oval top and bottom joined by two straight sides. Turtle Blocks has an arc block that turns while it moves. Four arcs make a flattened circle (an oval):

repeat 2
   arc 90, radius 20
   arc 90, radius 60

Draw one oval for the top, draw two straight lines down from its left and right edges, and draw another oval (or its front half) at the bottom. You can colour each face a different shade to make the depth even clearer.

Try it yourself

1. Each inside angle of a regular pentagon is 108°. How far must the turtle turn at each corner?
2. Fix Ishimwe's triangle program.
3. In the right-angled triangle program, what do the three turns add up to?
4. How many slanting lines join the front square and the back square of a cube?
5. Which shape has a front rectangle, a back rectangle and four slanting lines joining them?

Answers

1. 180 − 108 = 72°.
2. repeat 3 [forward 100, right 120].
3. 90 + 135 + 135 = 360°.
4. 4, one for each corner.
5. A cuboid.