Start with the drawing

Mark distinct points A and B on a circle. They divide it into two arcs, both including A and B. The chord AB is the straight segment between the points. An arc is not a segment.

If the extremes coincide, we distinguish the null arc, which is reduced to the point, and the arc of a turn, which covers the entire circumference. This distinction will be important when comparing the final position with the turn performed.

What is an angle?

A semi-straight has an origin and follows indefinitely in one direction. Two rays with origin O determine the two angular regions of the plane. When we use the smallest aperture, we need to indicate this; the opening on the other side also exists.

If the rays are opposite, each region corresponds to a shallow angle of $180°$. If they coincide, we can consider the null angle of $0°$ or that of a turn of $360°$.

The same opening, different arches

Draw two circles with the same center, one with radius 2 cm and the other with radius 4 cm. Two semi-straight lines leaving the center cut arcs in both. The central angle is the same, but the arc of the larger circle is twice as long.

The ratio between the radii is:

$$\frac42=2.$$

An enlargement of factor 2 doubles all lengths and preserves the angles. Therefore, the angular measure of an arc and its length are different information.

Worked example

Two rays delimit a quarter turn. The central angle measures:

$$\alpha=\frac{360°}{4}=90°.$$

On any circle with this center, the chosen arc occupies a quarter of the circle. Its angular measure is $90°$, even though the arc length changes with the radius.

In everyday life

  • Architecture. The arch over old doors and windows is a circumferential arc; the straight beam that crosses it is a rope.
  • Compass. The opening of the compass is an angle, and the line it makes is an arc.
  • Fan. The blades of a fan are at equal angles around the axis: three blades, $120^\circ$ between them.
An arched window: the top part is a circular arch, and the beam that crosses it is a rope.

Exercises with resolution

1. The points of a chord are the ends of how many arcs when they are distinct?

Check resolution 1

Two arches. One may be smaller and the other larger; if the points are extremes of a diameter, the two are semicircles.

2. A shallow angle corresponds to what fraction of a turn?

Check resolution 2

$180°/360°=1/2$, or half turn.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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