Understanding the Arc Length Formula

Arcs are vital elements in fields like geometry, physics, trigonometry, and design. Curved lines, however, are more challenging to measure than straight ones, which is why it's essential to grasp the arc length formula.

To calculate arc length, it's crucial to understand that an arc represents a portion of a circle, and its length is directly proportional to the fraction of the circle’s circumference. These calculations are key when designing bridges, plotting rocket paths, or creating roller coasters.

Continue reading to discover how to calculate the arc length and other key values like the radius and central angle.

What Is the Arc Length Formula?

The arc length formula is Arc Length = Radian Angle x Arc Radius or Arc Length = (Degree Angle/360) x 2π x Arc Radius, depending on whether you're working with radians or degrees.

If you don't have the arc radius or are unsure about the role of pi in this equation, be sure to keep reading for more information.

Calculating Circumference

Before we can calculate arc length, it's essential to first understand how to determine circumference and the relationship between the circumference and a circle's (or arc's) radius. This concept, discovered by the ancient Greeks over 2,000 years ago, was symbolized by the Greek letter π (pi).

If we draw a straight line across the center of a circle on a piece of paper, we create what's called a diameter. The straight-line distance from the center to the edge is known as the radius, which is always half the length of the diameter.

The Discovery of π

The Greeks also discovered that by stretching a string around the edge of a circle, the circumference (the distance around the circle) was just a bit more than three times the length of the string used for the diameter.

They named this value π (Pi) and refined the calculation of this ratio for more accurate measurements.

The true value of π involves an infinite series of non-repeating decimal digits, but for practical purposes, we use 3.14159. The exact value is not crucial, as most calculators feature a π key with a more accurate value.

With this in mind, the formula for calculating the circumference of any circle is the diameter multiplied by π (Dπ). You might also see it written as two times the radius times π (2rπ), but the outcome will be identical with either version of the formula.

For example, if a circle has a diameter of 4 inches, the circumference would be calculated as follows:

Alternatively, since the radius is half the diameter (4 in / 2 = 2 in), we can calculate using the radius:

Taking an Arc From a Circle

As previously mentioned, an arc represents a portion of a circle, and an arc shares the same radius as its corresponding circle. These two key points are essential when calculating the arc length.

If we know the curvature of an arc and the position of its center point, we can determine the arc length, though tools like a ruler and protractor may be necessary.

Measure the Radius

To find the radius, draw two line segments from the edge of the arc to its center. Use a ruler to measure the straight-line distance from the arc to the center point. This distance is the radius.

We can apply the circumference formula from earlier to calculate what the full circle’s circumference would be, but first, we need to determine the fraction of the circle that the arc represents. To do so, we'll need the angle value.

Finding the Central Angle

The central angle is the angle formed when the radii line segments from the previous step meet at the center, and this angle is the last piece of information required to calculate the arc length.

If your instructor or textbook hasn't provided the central angle, you can draw two radii at the arc's edges and use a protractor to measure the angle formed at the center.

You can measure the angle in either degrees or radians. If the central angle exceeds 180 degrees (or π radians), you may choose to measure the opposite angle and express it as the given angle plus 180 degrees.

Degrees vs. Radians

Although both degrees and radians measure the same thing, they serve slightly different purposes. A degree represents 1/360th of a circle, while a radian is derived from the ratio of the radius to the circumference. Thus, one radian equals 1/2π.

A full circle is always 2π radians because, as you may recall, circumference is equal to 2π times the radius of the circle. We also have two handy formulas that can be used 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 has a close relationship with arc length, we have just about everything we need to calculate the arc length.

To better grasp how fractions of a circle work, think of a 90-degree angle (or 1/2 π radians) as one-quarter of a circle, 180 degrees (π radians) as half a circle, and 360 degrees (2 π radians) as a complete circle. In real-world scenarios, you’ll usually encounter fractional angles that fall somewhere between these standard ones.

A key concept to remember is the Arc Length Formula.

After determining the central angle, you can substitute it into the previously mentioned circumference formula to derive an arc length formula. To create the arc length formula for degrees, simply divide your central angle by 360, then multiply by the circumference formula.

When calculating the arc length using radians, the process becomes more straightforward. Since a radian is already a portion of 2π, and we need to multiply by 2π to determine the arc length, the 2π terms cancel each other out, simplifying the formula.

Calculating the Arc Length

We can now apply all the information we've gathered to work through a real example. Take the circle from our earlier illustration, which has a 4-inch diameter (2-inch radius), and let’s convert it into a fractional arc. We’ve measured the central angle (or arc) to be 136 degrees (2.374 radians).

At this point, we have everything we need to calculate the arc length!

First, let’s apply the arc length formula for degrees: (Degree Angle/360) x 2π x Arc Radius

Next, we can repeat the same calculation but this time using radians for the central angle.

As demonstrated, finding the arc length is straightforward as long as you have the radius and central angle. Our arc length formula can be applied consistently for both degrees and radians, yielding the same arc length in both cases.