How to Use the Calculator?
To use the Arc Length Calculator, simply enter any two known values, such as the central angle and radius, or the radius and arc length. The calculator will automatically work out the remaining values for you. You can choose the appropriate unit for each measurement using the unit dropdowns.
What Is an Arc?
Suppose two points, A and B, lie on the circumference of a circle. The curved part of the circle connecting A and B is an arc.
The size of an arc is closely related to the angle formed at the center of the circle. If we draw radii from the center O to points A and B, they form an angle called the central angle \(\theta\). A small central angle cuts off a short arc, while a larger central angle cuts off a longer arc.
Minor Arcs and Major Arcs
Two points on a circle normally divide its circumference into two arcs. The shorter path between the points is called the minor arc, while the longer path around the circle is called the major arc.
A minor arc corresponds to a central angle smaller than \(180^\circ\). A major arc corresponds to an angle greater than \(180^\circ\).
If the two points are opposite ends of a diameter, each arc is exactly half of the circle and corresponds to an angle of \(180^\circ\). Such an arc is called a semicircle.
Arc Length Formula
Sometimes we want to know how long the arc is. The distance measured along the arc is called its arc length.
Before finding the length of an arc, recall that the circumference of a complete circle is:
\[ C=2\pi r \]
where \(r\) is the radius of the circle.
A complete circle represents an angle of \(360^\circ\). Therefore, an arc with a central angle of \(\theta\) degrees represents the fraction
\[ \frac{\theta}{360} \]
of the complete circle.
The arc length is the same fraction of the circle’s circumference. Therefore:
\[ \text{Arc Length}=\frac{\theta}{360}\times 2\pi r \]
If we use \(s\) to represent arc length, the formula can be written as:
\[ s=\frac{\theta}{360}\times 2\pi r \]
Here, \(s\) is the arc length, \(\theta\) is the central angle measured in degrees, and \(r\) is the radius of the circle.
Example 1: Finding the Length of a Quarter Circle
A circle has a radius of 8 cm. Find the length of an arc with a central angle of \(90^\circ\).
Start with the arc-length formula:
\[ s=\frac{\theta}{360}\times2\pi r \]
Substitute \(\theta=90^\circ\) and \(r=8\):
\[ s=\frac{90}{360}\times2\pi(8) \]
Since \(\frac{90}{360}=\frac14\):
\[ s=\frac14\times16\pi \]
Therefore:
\[ s=4\pi\text{ cm} \]
Using \(\pi\approx3.14\):
\[ s\approx12.57\text{ cm} \]
So the arc is approximately 12.57 cm long.
Example 2: Finding an Arc Length
A circle has a radius of 12 cm and a central angle of \(120^\circ\). Find the length of the arc.
Using the formula:
\[ s=\frac{120}{360}\times2\pi(12) \]
Simplify the fraction:
\[ s=\frac13\times24\pi \]
Therefore:
\[ s=8\pi\text{ cm} \]
Using \(\pi\approx3.14\):
\[ s\approx25.13\text{ cm} \]
So the arc length is approximately 25.13 cm.
Example 3: Arc Length of a Semicircle
A circle has a radius of 7 cm. Find the length of the curved part of a semicircle.
A semicircle has a central angle of \(180^\circ\), so:
\[ s=\frac{180}{360}\times2\pi(7) \]
Since \(\frac{180}{360}=\frac12\):
\[ s=\frac12\times14\pi \]
Therefore:
\[ s=7\pi\text{ cm} \]
Using \(\pi\approx3.14\):
\[ s\approx21.99\text{ cm} \]
The curved portion of the semicircle is therefore approximately 21.99 cm long.
Example 4: Finding the Length of a Major Arc
A circle has a radius of 6 cm. Two radii form a \(100^\circ\) angle. Find the length of the major arc between the same two points.
The \(100^\circ\) angle describes the minor arc. Since a complete circle contains \(360^\circ\), the angle corresponding to the major arc is:
\[ 360^\circ-100^\circ=260^\circ \]
We can now use \(260^\circ\) in the arc-length formula:
\[ s=\frac{260}{360}\times2\pi(6) \]
Simplifying:
\[ s=\frac{13}{18}\times12\pi \]
\[ s=\frac{26\pi}{3}\text{ cm} \]
Using \(\pi\approx3.14\):
\[ s\approx27.23\text{ cm} \]
Therefore, the major arc is approximately 27.23 cm long.
Example 5: Finding the Central Angle from the Arc Length
Sometimes the arc length and radius are given, and the unknown quantity is the central angle. We still begin with the same formula:
\[ s=\frac{\theta}{360}\times2\pi r \]
We can rearrange it to find \(\theta\):
\[ \theta=\frac{360s}{2\pi r} \]
This can also be simplified to:
\[ \theta=\frac{180s}{\pi r} \]
For example, suppose an arc has a length of \(5\pi\) cm in a circle with a radius of 10 cm. Find the central angle.
Start with:
\[ s=\frac{\theta}{360}\times2\pi r \]
Substitute \(s=5\pi\) and \(r=10\):
\[ 5\pi=\frac{\theta}{360}\times20\pi \]
Divide both sides by \(20\pi\):
\[ \frac14=\frac{\theta}{360} \]
Multiply both sides by 360:
\[ \theta=90^\circ \]
Therefore, the central angle is \(90^\circ\).
Example 6: Finding the Radius from the Arc Length
The same formula can also be used when the radius is unknown. If the arc length and central angle are known, substitute them into the formula and solve for \(r\).
For example, suppose an arc has a length of \(6\pi\) cm and a central angle of \(120^\circ\). We have:
\[ 6\pi=\frac{120}{360}\times2\pi r \]
Since \(\frac{120}{360}=\frac13\):
\[ 6\pi=\frac13\times2\pi r \]
\[ 6\pi=\frac{2\pi r}{3} \]
Multiplying both sides by 3 gives:
\[ 18\pi=2\pi r \]
Dividing by \(2\pi\):
\[ r=9\text{ cm} \]
Therefore, the radius of the circle is 9 cm.
Arc Length and Sector Area
Arc length should not be confused with the area of a sector. An arc length measures a curved distance along the circumference of a circle, so its answer is given in ordinary length units such as centimeters, meters, or inches.
A sector, on the other hand, is the region enclosed by two radii and the arc between them. Because sector area measures the amount of surface inside that region, it is measured in square units such as \(\text{cm}^2\) or \(\text{m}^2\).
For arc length, we use:
\[ s=\frac{\theta}{360}\times2\pi r \]
For sector area, the corresponding formula is:
\[ A=\frac{\theta}{360}\times\pi r^2 \]
The two formulas look similar because both compare the central angle with \(360^\circ\). The important difference is that arc length uses the circumference \(2\pi r\), while sector area uses the area of the circle \(\pi r^2\).
