Arc Length Calculator

The Arc Length Calculator is a fast and accurate online tool for finding the length of an arc from different known circle values. Whether you know the radius and central angle, chord length and segment height, or other related measurements, this calculator simplifies the process and returns the result instantly.

In addition to arc length, it also provides useful outputs such as diameter, chord length, sector area, triangle area, and segment area, along with a clear step-by-step explanation. This makes it ideal for geometry students, teachers, engineers, architects, and professionals who need reliable circle calculations for study, design, or technical work.

What is Arc Length?

A circle's boundary is known as its circumference. If you take just a small part of that edge, it is called an arc. The length of this curved part is known as the arc length. An arc length is determined by a radius and a central angle. The radius is the distance from the center of the circle to its circumference, and the central angle is the angle made with a vertex at the center of the circle by the two lines that extend to the ends of the arc. 

Arc Length Equation:

To calculate the arc length manually, you must use the correct equation. The arc length formula changes slightly depending on whether you measure the central angle in radians or degrees.

If the central angle is in radians:

s = rθ

If the central angle is in degrees:

s = θ/360 * 2πr

s = (𝜃 / 360) × 2πr

Solved Example

In order to remove any complication, we will provide a practical example illustrating how to determine arc lengths, as guided by the top formula of length of an arc.

The Problem:

Imagine you are building a circular garden path. You know the radius of the circle is 10 feet. The path covers a central angle of 45 degrees. What is the length of arc for this specific path section?

Step 1. Identify values

Radius (r) = 10 feet.

Central angle (𝜃) = 45°

Step 2. Select the appropriate arc length formula

Since the angle is in degrees, we will use the following:

s = (𝜃 / 360) × 2πr

Step 3. Insert the values

s = (45 / 360) × 2 × 3.14159 × 10

Step 4. Solve the equation.

First, divide the angle by 360.

45 / 360 = 0.125

Then, determine the full circumference: 2 × 3.14159 × 10 = 62.8318 feet

Finally, multiply the two results: 0.125 × 62.8318 = 7.8539 feet

Thus, the length of the path taken is approximately 7.85 feet. This can be computed by hand. However, the arc length calculator can do it in milliseconds.

Arc Length and Sector Area Relationship

Arc length and radius of the sector are two different measures that describe the same circular slice. This slice can be compared to a slice of pizza.

With the pizza example:

• The arc length would be the length of the pizza crust.
• The sector would be the whole slice, including crust, cheese, toppings, etc.

Where the arc length and radius of the sector measure depend on

• The radius (how far the center is from the edge)
• The angle (wideness of the slice).

When measuring angles in radians, the formula for measuring the sector radius is:

\[A = \frac{1}{2} r^2 \theta\]

At the same time, the formula for measuring the arc length is:

\[s = r \theta\]

If we denote arc length as \(s\), the two equations can be combined into one simplified formula:

\[A = \frac{1}{2} r s\]

This also shows that as long as you have the arc length (s) and radius (r), you can easily find the sector area without directly measuring the angle. Hence, the radius of the sector and the arc length are directly correlated.

Frequently Asked Questions

Can I trust this tool for engineering work?

Yes. It uses the exact values of 𝜋 and floating-point calculations for the perfect output in CAD, engineering, and architecture.

Can I use a formula to find arc length without a central angle?

The central angle is crucial. Without it, the problem is unclear, and it indicates how much of the circle's total circumference to use.

Can the Countivia calculator differentiate between degrees and radians?

Yes, it does. It allows you to choose degrees or radians, and the appropriate formula of the arc length will be used.

Is the Countivia arc length calculator free?

Yes! You can use the calculator for any work-related assignments or just personal interests, as many times as you want, without any fees.

Is radius conversion necessary?

No. The calculator is informal about the radius, and as a result, your answer will be in the same unit of measure as your radius.

Is this calculator mobile-friendly?

Yes. The calculator is mobile-friendly and is responsive across devices from mobile to tablet to desktop.

Do I need to convert the radius to a certain measurement?

Not at all. The calculator does not require a certain unit of measurement. If you measure in inches, the answer will also be in inches.

How do I reset the calculator for a new problem?

You would either delete values that are already in the input boxes or completely start a new calculation and reset the calculator. You can just refresh the page.

Reference Studies

Khan Academy, Arc length from subtended angle.

OpenStax, 7.1 Angles – Algebra and Trigonometry 2e.

Mathematics LibreTexts, 2.2: Applications of Radian Measure.

Louisiana State University Department of Mathematics, Section 6.2 Applications of Radian Measure.

Texas A&M University Open Digital Publishing, 7.1 Degree and Radian Measure of Angles – Functions, Trigonometry, and Systems of Equations (Second Edition)