Euler's Eyes
2026-10-21
Corners, spokes and paths on the wheel
Two paths between the same points
Draw a wheel and mark two points on it, A and B. Connect them with a straight line: it is a rope. Now follow the edge of the wheel from A to B: you have covered an arc. There are two possible paths around the edge; indicate which one you are using.
The center is the point in the middle of the wheel. A segment from the center to the edge is a radius. All spokes on the same wheel have the same length.
A song we can carry
Fold a sheet in half and then make another fold perpendicular to the first. The corner formed is a right angle model, which measures $90°$. Place this corner against doors, tables and drawings: which openings are the same as it?
Two neighboring right angles form a half-turn:
$$90°+90°=180°.$$Four complete the lap:
$$90°+90°+90°+90°=360°.$$Why doesn’t arm length change singing?
Draw two arrows coming from the same point. Extend the two arrows without changing their directions. The opening remains the same. Thus, an angle depends on the directions, not the size of your arms design.
A larger wheel may have a longer arc for the same spin. $90°$’s corner remains the same: he occupies a quarter of the large wheel and also a quarter of the small one.
In everyday life
- Door. An open door forms a corner with the wall; the opening is the same for the small closet door and the large living room door.
- Pizza. All slices of a pizza cut into equal parts have the same corner in the center, even if the pizza is large or small.
- Scissors. Opening the scissors more or less changes the corner between the blades, not their size.
Try it
Use two pencils as arms of an angle. Keep the ends together. Open the pencils more or less and compare with the corner of the page. Describe each opening as less than, equal to, or greater than a right angle.
Exercises with resolution
1. A wheel has a radius of 3 cm. How long is another spoke on the same wheel?
Check resolution 1
Also 3 cm. The equality of the radii is part of the definition of a circumference.
2. Are an edge path and a straight path between two points given the same name?
Check resolution 2
No. The path around the edge is an arch; the straight segment is a chord.
Correspondence with source material
Preparatory activity on angles and triangles, adapted for this stage. General formulas appear in the following tracks.
The text of this class is a didactic rewrite. The page transcription preserves the sequence of exercises and other items from the original; it still has OCR snippets pending review.