Length of an Arc Formula: Easy Guide with Solved Examples
Updated on April 28, 2026
arc length is the distance between two points along a curved line or segment of a circle. It represents a portion of the total circumference of a circle, measured in linear units such as centimeters, meters, or inches. Understanding how to calculate this distance is essential in fields ranging from basic geometry to advanced physics and engineering. Students seeking additional support can explore personalized guidance through geometry tutor.
The length of an arc depends directly on the radius of the circle and the size of the central angle that the arc subtends. A central angle is an angle whose vertex is at the center of the circle and whose sides are radii that meet the circle at the endpoints of the arc. As the central angle increases, the portion of the circumference it captures also grows, resulting in a longer arc length.
There are different formulas used to find arc length depending on whether the central angle is measured in degrees or radians. While the concept remains the same—finding a fraction of the total circumference—the units of the angle dictate the specific mathematical approach. Mastering these formulas allows students to solve real-world problems involving circular motion, architecture, and design.
What is length of an arc formula?
The length of an arc formula is a mathematical expression used to determine the distance along a curved path of a circle by relating the central angle to the total circumference. In its most basic form, it calculates a specific fraction of the circle’s perimeter based on the angle formed at the center. For a full circle, the angle is 360 degrees or 2π radians, and the arc length is equal to the circumference; for any smaller angle, the formula provides the corresponding partial distance.
How to Find Arc Length With Radius and Central Angle
To find the arc length using the radius and central angle, you must identify the total circumference of the circle and then determine what fraction of that circumference is covered by the arc. The radius is the distance from the center to any point on the edge, and the central angle defines the “slice” of the circle you are measuring. The calculation involves multiplying the circumference by the ratio of the central angle to the full 360-degree or 2π-radian rotation of the circle.
Arc Length Formula in Radians
When the central angle is measured in radians, the arc length formula is exceptionally simple because of how radians are defined. A radian is the angle created when the arc length is exactly equal to the radius. Therefore, the arc length (s) is simply the product of the radius (r) and the central angle (θ) in radians. The formula is expressed as:
- s = r × θ
This formula is preferred in advanced mathematics and physics because it eliminates the need for constant conversion factors. Since a full circle is 2π radians and the circumference is 2πr, the relationship remains direct and proportional. To use this formula, always ensure the angle is not in degrees; if it is, you must convert it by multiplying the degree measure by π/180 before proceeding with the calculation.
Arc Length Formula in Degrees
In many K-12 math problems, the central angle is provided in degrees. Because a full circle consists of 360 degrees, the arc represents a fraction of the circle equal to θ/360. To find the arc length, you multiply the circumference of the circle (2πr or πd) by this fraction. The formula for arc length when the angle is in degrees is:
- s = 2πr × (θ / 360°)
- s = πd × (θ / 360°)
This can be simplified to s = (πrθ) / 180°. In this equation, s is the arc length, r is the radius, and θ is the central angle in degrees. This version of the formula is highly practical for everyday measurements, such as determining the length of a curved piece of track or the edge of a circular table, where protractors measuring in degrees are the standard tool for observation.
General Arc Length Formula for Functions
The general arc length formula for functions is used in calculus to find the distance along a smooth curve defined by a function y = f(x) over a specific interval. Unlike a circle, where the curvature is constant, the curvature of a function can change at every point. To find the total length, the curve is divided into infinitely small straight segments, and the lengths of these segments are summed using an integral. This process is known as rectification of the curve.
For a function y = f(x) that is continuous and has a continuous derivative on the interval [a, b], the formula for the arc length (s) is the integral from a to b of the square root of one plus the square of the derivative of the function. This is written as:
- s = ∫[a, b] √[1 + (dy/dx)²] dx
If the curve is defined parametrically by x = x(t) and y = y(t) over the interval [a, b], the formula becomes the integral of the square root of the sum of the squares of the derivatives with respect to t. This advanced application allows for the measurement of complex paths, such as the trajectory of a projectile or the shape of a suspension cable, which do not follow simple circular arcs.
Solved Examples on length of an arc formula
Applying the arc length formula requires identifying the given variables—radius, diameter, or angle—and selecting the appropriate version of the formula based on the units provided. These examples demonstrate the step-by-step process for solving common geometry problems involving circular arcs. By following a consistent method of substitution and calculation, students can accurately determine the linear distance of any curved segment within a circle.
Example 1
Problem: Find the arc length of a circle with a radius of 10 centimeters and a central angle of 60 degrees. Round the answer to two decimal places.
Solution: First, identify the variables: r = 10 and θ = 60°. Since the angle is in degrees, use the degree formula s = 2πr × (θ / 360°). Substitute the values into the equation: s = 2 × 3.14159 × 10 × (60 / 360). This simplifies to s = 62.8318 × (1 / 6). Calculating the final result gives s = 10.47 centimeters. The distance along the curve for a 60-degree segment of this circle is approximately 10.47 cm.
Example 2
Problem: A circular garden has a radius of 5 meters. An arc-shaped walkway covers a central angle of 2 radians. Calculate the exact length of the walkway.
Solution: Identify the variables: r = 5 and θ = 2 radians. Because the angle is already in radians, use the simpler radian formula s = r × θ. Substitute the values: s = 5 × 2. The calculation results in s = 10. The length of the arc-shaped walkway is exactly 10 meters. This example illustrates how radian measure significantly simplifies the calculation by removing the need for π or the 360-degree conversion factor.
Example 3
Problem: Determine the central angle in degrees if the arc length is 15.7 inches and the radius of the circle is 6 inches. Use 3.14 for π.
Solution: Identify the knowns: s = 15.7 and r = 6. Use the degree formula and solve for θ: 15.7 = 2 × 3.14 × 6 × (θ / 360). Simplify the constants: 15.7 = 37.68 × (θ / 360). Divide both sides by 37.68: 0.4166 = θ / 360. Multiply by 360 to find θ: θ = 0.4166 × 360, which equals approximately 150 degrees. The central angle that creates an arc of 15.7 inches in this circle is 150°.
Example 4
Problem: Calculate the arc length of a semicircle with a diameter of 14 feet. Give the answer in terms of π.
Solution: Identify the variables: d = 14, so r = 7. A semicircle has a central angle of 180 degrees. Using the degree formula: s = 2π(7) × (180 / 360). This simplifies to s = 14π × (1 / 2). The final result is s = 7π feet. Since 180 degrees is exactly half of a circle, the arc length of a semicircle is always half of the total circumference (πr), which matches our result of 7π.
FAQ
How do you find the arc length of a circle?
To find the arc length of a circle, you must first determine the radius of the circle and the measure of the central angle that forms the arc. Once you have these values, check the units of the angle. If the angle is in degrees, use the formula s = 2πr(θ/360), which calculates the arc as a fraction of the total circumference. If the angle is in radians, use the formula s = rθ. Always ensure that your radius and arc length are in the same linear units, such as inches or centimeters, to maintain consistency in your final answer.
What is the difference between arc length and sector area?
Arc length and sector area are related but measure different properties of a circle. Arc length is a linear measurement that describes the distance along the curved edge of a “slice” of the circle, similar to measuring the crust of a pizza slice. Sector area is a two-dimensional measurement that describes the total surface space enclosed within the arc and the two radii, similar to measuring the entire surface of that pizza slice. While arc length uses formulas like s = rθ, sector area uses formulas like A = 0.5r²θ, resulting in units that are squared.
Is arc length measured in square units?
No, arc length is never measured in square units. Because an arc is a segment of a line—even though that line is curved—it represents a one-dimensional distance. Therefore, arc length is measured in linear units such as millimeters (mm), centimeters (cm), meters (m), inches (in), or feet (ft). Square units, such as square centimeters or square inches, are reserved for measuring area, which describes a two-dimensional surface. When solving problems, always label your arc length answer with the same units provided for the radius or diameter of the circle to ensure accuracy.
Can you find arc length without the radius?
You cannot find the arc length with only the central angle; you must have a measure of the circle’s size, such as the radius, diameter, or total circumference. However, if you are given the sector area and the central angle, you can calculate the radius first and then find the arc length. Similarly, if you know the chord length and the central angle, you can use trigonometry to find the radius. While the radius is a core component of the formula, it can be derived from other geometric properties of the circle if it is not directly provided.
What is the formula for arc length if the angle is in radians?
If the central angle is measured in radians, the formula for arc length is s = rθ. In this equation, s represents the arc length, r is the radius of the circle, and θ is the central angle in radians. This formula is derived from the very definition of a radian: one radian is the angle where the arc length equals the radius. Because of this direct relationship, no additional conversion factors like π or 360 are needed. This simplicity makes radians the preferred unit for arc length calculations in physics, engineering, and calculus applications where efficiency is important.
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