Breaking down the arc length formula

Arcs are an important aspect of geometry, physics, trigonometry and design work. However, curved lines are much more difficult to measure than straight lines. Therefore, it is important to familiarize yourself with the formula for arc length.

The key to calculating arc length is to understand that an arc is fundamentally a fraction of a whole circle, and that the corresponding length is an equal part of the circle’s circumference. These types of calculations are essential if you want to design a bridge, project the trajectory of a rocket or build a roller coaster.

Read on to learn how to calculate arc length, as well as other important values ​​such as radius and central angle.

What is the arc length formula?

The formula for arc length is Arc length = Radial angle x Arc radius or Arc length = (angle/360 degrees) x 2π x arc radiusdepending on whether you measure in radians or degrees.

That said, if you don’t already have the radius of arc or know the role of pi in this situation, you’ll want to keep scrolling for more details.

Calculate circumference

Before we can calculate arc length, we must first understand how to calculate circumference, as well as the relationship between the circumference and the radius of a circle (or arc). It is said that this relationship was discovered more than 2,000 years ago by the ancient Greeks, who assigned the Greek letter π (pi) to describe it.

If we have a circle on a piece of paper and draw a line segment straight through the center and to the other side, we create what is known as a diameter. The straight line distance directly to the center is called the Raywhich is always exactly half the length of the diameter.

find π

The Greeks also discovered that, by making a diameter line segment from a piece of string and wrapping it around the curved line edge of the circle, the circumference (or length around the circle) was equal to just over three times the chord length of their string. .

They named this value π (Pi) and calculated more specific values ​​for this ratio.

The actual value of π uses an infinite, non-repeating number of decimal digits, but for our purposes we can say that π is equal to 3.14159. The exact value of the number is not that important, as most calculators include a π key with a more precise number.

With all that in mind, the formula for calculating the circumference of a circle is the length of the diameter of the chord length times π (Dπ). You may also see this expressed as two times the radius times π (2rπ), but the result will be exactly the same for either version of the formula.

For example, if a circle has a diameter of 10 cm, its circumference is:

or, since we know the radius is half the diameter (4 in / 2 = 2 in):

Taking an arc from a circle

As mentioned, an arc is a fraction of a circle, and an arc has the same radius length as the corresponding one. These two facts are the most important to keep in mind when trying to calculate arc length.

As long as the curve of an arc is known and the location of the center is known, we can find the arc length, although additional tools such as a ruler and a protractor may be needed.

Measure the radius

If we know the center of our arc, we can find the length of the radius by drawing two line segments from the edge of our arc to the center. Use a ruler to measure the straight line distance between the arc and the center, and that is our radius.

We can use our perimeter conversion formula above to see what the perimeter length would be if our arc were a complete circle, but first we’ll need to figure out what the fractional value of the arc is. To do that, we need an angle value.

Finding the central corner

The central angle is the angle formed by drawing the radius line segments we created in the last step, and this angle measurement is the final value needed to find the arc length.

If your textbook or instructor has not indicated a center angle, you will need to draw two rays at the edges of the arc and then use a protractor to measure the angle where they meet at the center.

You can make this measurement in degrees or radians. If the central angle is greater than 180 degrees (π radians), you may want to measure the opposite side of the angle and then express it as your measurement plus 180 degrees.

Degrees vs radians

Although degrees and radians are units used to measure the same thing, they serve slightly different purposes. One degree represents 1/360 of a circle, while a radian is a function of the ratio of radius to circumference. For this reason, a radian is equal to 1/2π.

A complete circle is always 2π radians, because, as you may recall, the circumference is 2π times the radius of the circle. We also have two handy formulas you can use if you ever need to convert degrees to radians, or vice versa.

Fractions of a circle

Now that we know that central angle can be used to express a fraction of a circle, and that radius is closely related to arc length, we have just about everything we need to calculate arc length.

To better visualize parts of a circle, consider an angle of 90 degrees (1/2 π radians) as a quarter circle, 180 degrees (π radians) as a semicircle, and 360 degrees (2 π radians) as full circles . Of course, real world examples will often be fractional values ​​lying between these neat angles.

Memorize the formula for arc length

Once you find the central angle, you can plug it into the perimeter formula from earlier to create an arc length formula. To obtain the arc length formula for degrees, take your central angle value, divide it by 360, and then multiply the result by our perimeter formula.

If you calculate arc length using radians, the process is a little easier. Since a radian value is a fraction of 2π, and we need to multiply it by 2π to calculate the arc length, these two cancel out. We are left with this formula:

Calculation of the arc length

Now we can use all this information to solve an actual problem. Let’s convert our circle from the previous example with a diameter of 4 inches (2 inches radius) into a fractional arc. We measured the central angle or arc and found it to be 136 degrees (2.374 radians).

Finally, this is all the information we need to calculate arc length!

First, let’s use the arc length formula for degrees: (Degree angle/360) x 2π x arc radius

Now we can do the same problem again using radians for our center angle.

As you can see, determining arc length is simple as long as the radius and central angle are known. Our arc length formula can be used reliably to convert both degrees and radians, and we get two arc lengths with the same value.

Original article: Breaking the arc length formula

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