Arc Definition in Geometry

Circles are simple, but they do have parts. One part is an arc, a snippet of the circle, a piece of its circumference. Arcs themselves come in types, like major arcs, semicircles, and minor arcs.

Table Of Contents

1. Circles and Circumference
2. Semicircles and Arcs
3. Identifying Arcs
4. Measuring Arcs

Circles and Circumference

A circle is the set of all points equidistant from a given point. Circumference is the distance around a circle.

[insert drawing of circle with curved arrow going around it to show circumference]

Circles can have angles created by two radii. These are central angles and are almost always designated using either their exact angle (or radian) measurement or the Greek letter theta, ϴ:

[insert circle drawing with central angle ϴ]

Circles can also have angles created by two chords (line segments with endpoints on the circle) with a common endpoint on the circle. These angles are called inscribed angles:

[insert drawing of inscribed angle]

Both central angles and inscribed angles create major and minor arcs.

Semicircles and Arcs

An arc is a portion of a circle that is less than the entire circle. Since that allows nearly all possible portions, mathematicians break down arcs like this:

1. Minor arc -- An arc measuring less than or equal to 180° or π radians
2. Semicircle -- An arc measuring exactly 180° or π radians, which excludes designating either part of the circle as major or minor
3. Major arc -- An arc measuring greater than or equal to 180° or π radians

[Insert two drawings -- one of a circle divided into a minor arc and major arc, the other drawing showing a circle divided into two semicircles]

Identifying Arcs

In a typical drawing of a circle, the reader understands the minor arc to be the one under discussion. In this drawing, we care about the minor arc identified by the central angle ϴ:

[insert drawing of circle with central angle ϴ creating an acute angle, with the minor arc of that angle emphasized]

Labeling a minor arc requires only its endpoints on the circle. Here is minor arc GO:

[insert drawing of circle with minor arc GO labeled and emphasized]

If you want the major arc, select and label both endpoints of the arc and a random point between. Here we have major arc FUN:

[insert drawing of circle with three points F, U, N on the major arc]

Arcs are usually identified in writing using their points (two for a minor arc, three for a major arc) and then drawing a tiny, short arc drawn over the letters:

Measuring Arcs

Arcs have two measurements:

1. Angle
2. Length

One way to measure an arc is by the central angle of the circle. This is the arc angle. You place a lowercase m in front of the written form for the arc, like this:

So you could write mFUN = 45°, and you would say, "The major arc FUN measures 45 degrees."

The other way to measure arcs is by their distance along the circumference of the circle. This is the arc length. To write arc length in words, you put a small l in front of the written form, like this:

[refer to Figure 2 emailed, and put graphic arc above lGO below]

So you could write lGO = 13.4 cm and you would say, "The length of arc GO is 13.4 centimeters."

Next Lesson:

How To Find Arc Length