# Central Angles of a Circle

My teacher said that the central angles of a circle are equal to the measure of the arc, but I don't understand on how this could possibly work.

Can someone please explain how this is possible?

Here's an example that should hopefully help build intuition for this concept.

Suppose you hear church bells tolling the hour, and so you check your watch to see what hour it is. The minute-hand (which will be pointing towards the 12) and hour-hand create a central-angle inside the cirlce of the clockface. The numbers $1$ through $12$ mark off measures of arc around the circle. If central angles didn't equal the measures of the arc, then the angle of the hour-hand wouldn't equal the measure of the hour. That is, clocks simply would not work.

The central angle of a circle is the angle between any two lines drawn from the center of the circle, to it's outer rim. Basically any angle created in the center of the circle (hence the name "central angle"). The circumference of a circle is equal to $2\pi r$ (where $r$ is the circle's radius). There are $2\pi$ radians in a circle, therefore the central angle in a full circle is equal to $2\pi$. In a unit circle, the radius is of length $r=1$, therefore the circumference of the circle (aka, the circle's arc length) is equal to $2\pi r = 2\pi =$ (the circle's central angle). Therefor the circle's central angle, is equal to it's arc length, in the case of a unit circle, where $r=1$.

• Sorry, I don't understand how this would help with the angles. – IHeartBunnies May 27 '14 at 1:54
• @IHeartBunnies - I've since elaborated. – Ephraim May 27 '14 at 2:05

If an arc subtends a central angle, then the length of the arc divided by the length of the radius of the circle equals the measure of the angle, in radians. For a circle of radius $1$, the angle measure is equal to the arc length.

If $a$ is the length of the arc that subtends the angle, $b$ is the angle measure (in radians), and $c$ is the radius, then $$a = bc$$

This is why mathematicians like to measure angles in units of radians rather than degrees. Once around the circle is 360 degrees but only $2\pi$ radians.

Moreover, your teacher is talking about a circle whose radius is $1$. I hope he or she remembered to mention that.

• What do you mean a radius of 1? He never mentioned this to me. – IHeartBunnies May 27 '14 at 2:00
• That's the kind of circle whose arcs are equal to their central angles. For example, take one quarter of a circle whose radius is $1$. The entire circumference is $2\pi$, so one quarter circumference is $\pi/2$. The central angle of this same arc is a right angle, which is $\pi/2$ radians. – David K May 27 '14 at 2:04
• Ok I understand that now. Thanks! – IHeartBunnies May 27 '14 at 2:04

Most of the answerers in this thread are trying to explain to you something about the length of an arc, but I think they may be missing the point. Your question refers to the measure of the arc, not the length of the arc. In secondary geometry (which I assume is the context of your question) arcs are measured in degrees: a semi-circle has a measure of 180°, a quarter-circle has a measure of 90°, etc. That is what your teacher means by "the measure of an arc is equal to the measure of its central angle" -- it is, more or less, the definition of "arc measure".

Of course arcs also have length, and one of the things you will probably be expected to do is to compute the length of an arc with a given measure in a circle of a given radius. For this, the basic idea is to first divide the degree measure of the arc by 360° (which tells you what fraction of a circle the arc consists of) and then multiply that by the circumference of the full circle (which of course is $2 \pi r$).

Different textbooks may use different notation for arc measure, but most high school geometry textbooks in the United States use $m \stackrel \frown {AB}$ for the measure of arc $\stackrel \frown {AB}$. There does not appear to be a notation for the length of $\stackrel \frown {AB}$.

There's no need to measure things in radians, although of course it makes things simpler (in some ways) if you do.