How to find an arc of a circle

Video transcript

I have a circle here whose circumference is 18 pi. So if we were to measure all the way around the circle, we would get 18 pi. And we also have a central angle here. So this is the center of the circle. And this central angle that I'm about to draw has a measure of 10 degrees. So this angle right over here is 10 degrees. And what I'm curious about is the length of the arc that subtends that central angle. So what is the length of what I just did in magenta? And one way to think about it, or actually maybe the way to think about it, is that the ratio of this arc length to the entire circumference-- let me write this down-- should be the same as the ratio of the central angle to the total number of angles if you were to go all the way around the circle-- so to 360 degrees. So let's just think about that. We know the circumference is 18 pi. We're looking for the arc length. I'm just going to call that a. a for arc length. That's what we're going to try to solve for. We know that the central angle is 10 degrees. So you have 10 degrees over 360 degrees. So we could simplify this by multiplying both sides by 18 pi. And we get that our arc length is equal to-- well, 10/360 is the same thing as 1/36. So it's equal to 1/36 times 18 pi, so it's 18 pi over 36, which is the same thing as pi/2. So this arc right over here is going to be pi/2, whatever units we're talking about, long. Now let's think about another scenario. Let's imagine the same circle. So it's the same circle here. Our circumference is still 18 pi. There are people having a conference behind me or something. That's why you might hear those mumbling voices. But this circumference is also 18 pi. But now I'm going to make the central angle an obtuse angle. So let's say we were to start right over here. This is one side of the angle. I'm going to go and make a 350 degree angle. So I'm going to go all the way around like that. So this right over here is a 350 degree angle. And now I'm curious about this arc that subtends this really huge angle. So now I want to figure out this arc length-- so all of this. I want to figure out this arc length, the arc that subtends this really obtuse angle right over here. Well, same exact logic-- the ratio between our arc length, a, and the circumference of the entire circle, 18 pi, should be the same as the ratio between our central angle that the arc subtends, so 350, over the total number of degrees in a circle, over 360. So multiply both sides by 18 pi. We get a is equal to-- this is 35 times 18 over 36 pi. 350 divided by 360 is 35/36. So this is 35 times 18 times pi over 36. Well both 36 and 18 are divisible by 18, so let's divide them both by 18. And so we are left with 35/2 pi. Let me just write it that way-- 35 pi over 2. Or, if you wanted to write it as a decimal, this would be 17.5 pi. Now does this makes sense? This right over here, this other arc length, when our central angle was 10 degrees, this had an arc length of 0.5 pi. So when you add these two together, this arc length and this arc length, 0.5 plus 17.5, you get to 18 pi, which was the circumference, which makes complete sense because if you add these angles, 10 degrees and 350 degrees, you get 360 degrees in a circle.

Arc Measure Definition

An arc is a segment of a circle around the circumference. An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. This angle measure can be in radians or degrees, and we can easily convert between each with the formula π radians =  180°.

You can also measure the circumference, or distance around, a circle. If you take less than the full length around a circle, bounded by two radii, you have an arc. That curved piece of the circle and the interior space is called a sector, like a slice of pizza. When you cut up a circular pizza, the crust gets divided into arcs.

1. Arc Measure Definition
2. Arc of a Circle
3. Arc Measure vs. Arc Length
4. Degrees and Radians
5. Arc Measure Formula
6. How To Find The Measure of an Arc
7. How To Find Arc Length
8. Identifying Arc Angle Indicated

Arc of a Circle

If we cut across a delicious, fresh pizza, we have two halves, and each half is an arc measuring 180 °. If we make three additional cuts in one side only (so we cut the half first into two quarters and then each quarter into two eighths), we have one side of the pizza with one big, 180° arc and the other side of the pizza with four, 45° arcs like this:

The half of the pizza that is one giant slice is a major arc since it measures 180° (or more). The other side of the pizza has four minor arcs since they each measure less than 180°.

Arc Measure vs. Arc Length

The arc is the fraction of the circle's circumference that lies between the two points on the circle. An arc has two measurements:

1. The arc's length is a distance along the circumference, measured in the same units as the radius, diameter or entire circumference of the circle; these units will be linear measures, like inches, cm, m, yards, and so on
   
2. The arc's angle measurement, taken at the center of the circle the arc is part of, is measured in degrees (or radians)
   

Do not confuse either arc measurement (length or angle) with the straight-line distance of a chord connecting the two points of the arc on the circle. The chord's length will always be shorter than the arc's length.

Degrees and Radians

To be able to calculate an arc measure, you need to understand angle measurements in both degrees and radians. An angle is measured in either degrees or radians. A circle measures 360 degrees, or 2π radians, whereas one radian equals 180 degrees. So degrees and radians are related by the following equations:

360° = 2π radians

180° = π radians

The relationship between radians and degrees allows us to convert to one another with simple formulas. To convert degrees to radians, we take the degree measure multiplied by pi divided by 180.

Let's convert 90 degrees into radians for example:

90° × π180°

90π180

π2 radians

Now let's convert π3 radians to degrees:

π3 × 180π

180π3π

1803 =  60°

Arc Measure Formula

Once you got the hang of radians, we can use the arc measure formula which requires the arc length, s, and the radius of the circle, r, to calculate.

arc m easure = arc lengthradius = sr

How To Find The Measure of an Arc

Let's try an example where our arc length is 3 cm, and our radius is 4 cm as seen in our illustration:

Start with our formula, and plug in everything we know:

arc me asure = sr

arc measure = 34

Now we can convert 34  radians into degrees by multiplying by 180 dividing by π.

34180π

42.9718  ≈ 43°

How To Find Arc Length

You need to know the measurement of the central angle that created the arc (the angle of the two radii) to calculate arc length. The arc length is the fractional amount of the circumference of the circle. The circumference of any circle is found with 2πr where r = radius. If you have the diameter, you can also use πd where d = diameter.

The formula for finding arc length is:

Arc length = arc angle360 ° 2πr

Let's try an example with this pizza:

Our pie has a diameter of 16 inches, giving a radius of 8 inches. We know the slice is 60°. So the formula for this particular pizza slice is:

      = 60°360° · 2·π·8

       = 16 · 16π

      ≈ 8.3775"

Identifying Arc Angle Indicated

An arc angle's measurement is shown as mAB⌢ where A and B are the two points on the circle creating the arc. The m means measurement, and the short curved line over the AB⌢ indicates we are referring to the arc. The two points derived from the central angle (the angle of the two radii emerging from the center point).

One important distinction between arc length and arc angle is that, for two circles of different diameters, same-angle sectors from each circle will not have the same arc length. Arc length changes with the radius or diameter of the circle (or pizza).

Lesson Summary

Now that you have eaten your way through this lesson, you can identify and define an arc and distinguish between major arcs and minor arcs. You are also able to measure an arc in linear units and degrees and use the correct symbol, mAB⌢ (where A and B are the two points on the circle), to show arc length.

Next Lesson:

Tangent to a Circle

What is the formula to find arc?

What is the arc of a circle?