Arc Length Calculator

Q: What is the arc length formula?

The arc length formula is s = r x theta, where s is the arc length, r is the radius, and theta is the central angle in radians. When the angle is in degrees, use s = (pi x r x theta) / 180. This formula derives from the relationship between an arc and the circumference of a circle.

Q: How do you find the radius from arc length and angle?

To find the radius from arc length and angle, use r = s/theta where s is the arc length and theta is the angle in radians. If the angle is in degrees, first convert to radians by multiplying by pi/180, then divide the arc length by this value.

Calculate arc length, radius, or central angle of a circle sector with interactive diagram, step-by-step solution, and support for degrees and radians.

Arc Length Calculator

Quick Examples:

r=5, θ=45° r=10, θ=90° r=3, θ=π/2 arc=7.854, θ=90° r=6, arc=9.42

Arc Length

Radius

Angle

Calculation Mode

Radius (r)

Central Angle (θ)

Angle Unit

Arc Length

Decimal Precision

Tip: Use the interactive diagram above to visualize your sector. The diagram updates as you enter values.

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About Arc Length Calculator

Welcome to the Arc Length Calculator, a comprehensive geometry tool that calculates arc length, radius, or central angle of a circle sector. With an interactive SVG diagram, step-by-step mathematical solutions, and support for both degrees and radians, this calculator helps students, engineers, and professionals solve arc-related problems quickly and accurately.

What is Arc Length?

Arc length is the distance along the curved path of an arc - a portion of a circle's circumference. It represents the actual length you would measure if you laid a string along the curved edge of a sector. Arc length is fundamental in geometry, trigonometry, and has numerous real-world applications in engineering, architecture, and physics.

The arc length depends on two factors: the radius of the circle and the central angle that the arc subtends. A larger radius or larger angle produces a longer arc.

Arc Length Formula

Arc Length (radians)

$$s = r \times \theta$$

When the angle is given in degrees:

Arc Length (degrees)

$$s = \frac{\pi \times r \times \theta}{180}$$

Where:

s = Arc length
r = Radius of the circle
$\theta$ = Central angle (in radians or degrees)
$\pi$ = Pi (approximately 3.14159)

Related Formulas

From the arc length formula, we can derive formulas for finding radius and angle:

Radius from Arc Length

$$r = \frac{s}{\theta}$$ (angle in radians)

Angle from Arc Length

$$\theta = \frac{s}{r}$$ (result in radians)

How to Use This Calculator

Select calculation mode: Choose what you want to find - Arc Length, Radius, or Angle.
Enter known values: Input the values you know. The calculator will disable the field for the value being calculated.
Select angle unit: Choose between degrees or radians for the angle input/output.
Calculate: Click the Calculate button to see the result along with all related values.
Review steps: Examine the step-by-step solution showing the mathematical process.

Understanding the Diagram

The interactive SVG diagram shows:

Purple arc: The arc whose length is being calculated
Dashed lines: The two radii forming the sector
Purple angle indicator: The central angle of the sector
Center dot: The center of the circle

The diagram updates in real-time as you change the input values, providing visual feedback for your calculations.

Converting Between Degrees and Radians

Understanding the relationship between degrees and radians is essential for arc length calculations:

To convert degrees to radians: Multiply by $\frac{\pi}{180}$
To convert radians to degrees: Multiply by $\frac{180}{\pi}$

Common conversions:

90° = $\frac{\pi}{2}$ radians
180° = $\pi$ radians
360° = $2\pi$ radians
1 radian ≈ 57.2958°

Applications of Arc Length

Engineering and Construction

Arc length calculations are essential in designing curved roads, railway tracks, bridges, and architectural elements like domes and arches. Engineers use these formulas to determine material requirements and ensure proper curvature.

Manufacturing

In manufacturing, arc length is used for designing gears, pulleys, cam mechanisms, and any circular or curved components. It helps determine belt lengths, gear tooth profiles, and circular saw blade specifications.

Navigation

Arc length is used to calculate distances on Earth's surface (which is approximately spherical). Great circle distances between two points are essentially arc lengths on a sphere.

Physics

In physics, arc length is crucial for analyzing rotational motion, pendulum movement, and circular trajectories. It connects angular measurements to linear distances traveled.

Computer Graphics

Arc length parameterization is important in computer graphics for rendering smooth curves, animating objects along paths, and creating fonts and vector graphics.

Arc Length vs. Other Measurements

Arc Length vs. Chord Length

The chord is the straight line connecting the two endpoints of an arc. Arc length is always greater than chord length (except for a degenerate arc of 0 length). The chord formula is:

Chord Length

$$c = 2r \sin\left(\frac{\theta}{2}\right)$$

Arc Length vs. Sector Area

While arc length measures the curved boundary, sector area measures the enclosed region. The sector area formula is:

Sector Area

$$A = \frac{1}{2}r^2\theta$$ (angle in radians)

Frequently Asked Questions

What is the arc length formula?

The arc length formula is s = r × θ, where s is the arc length, r is the radius, and θ is the central angle in radians. When the angle is in degrees, use s = (π × r × θ) / 180. This formula derives from the relationship between an arc and the circumference of a circle.

How do you convert degrees to radians?

To convert degrees to radians, multiply the angle in degrees by π/180. For example, 90 degrees equals π/2 radians (approximately 1.5708). The formula is: radians = degrees × (π/180). Conversely, to convert radians to degrees, multiply by 180/π.

What is the difference between arc length and chord length?

Arc length is the distance along the curved path of the arc, measured along the circumference. Chord length is the straight-line distance between the two endpoints of the arc. Arc length is always greater than or equal to chord length, with equality only when the arc degenerates to a point.

How do you find the radius from arc length and angle?

To find the radius from arc length and angle, use r = s/θ where s is the arc length and θ is the angle in radians. If the angle is in degrees, first convert to radians by multiplying by π/180, then divide the arc length by this value.

What is a sector of a circle?

A sector is a region of a circle bounded by two radii and the arc between them, like a slice of pizza. The central angle determines the size of the sector. A semicircle (180 degrees) and quadrant (90 degrees) are special cases of sectors.

How is arc length used in real life?

Arc length calculations are used in engineering for curved roads and railways, architecture for domes and arches, manufacturing for circular parts and gears, navigation for calculating distances on Earth's surface, and physics for analyzing rotational motion and pendulums.

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Additional Resources

Reference this content, page, or tool as:

"Arc Length Calculator" at https://MiniWebtool.com/arc-length-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 15, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Arc Length Calculator

About Arc Length Calculator

What is Arc Length?

Arc Length Formula

Related Formulas

How to Use This Calculator

Understanding the Diagram

Converting Between Degrees and Radians

Applications of Arc Length

Engineering and Construction

Manufacturing

Navigation

Physics

Computer Graphics

Arc Length vs. Other Measurements

Arc Length vs. Chord Length

Arc Length vs. Sector Area

Frequently Asked Questions

What is the arc length formula?

How do you convert degrees to radians?

What is the difference between arc length and chord length?

How do you find the radius from arc length and angle?

What is a sector of a circle?

How is arc length used in real life?

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