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Arc Length Calculator
Easily calculate arc length, sector area, and visualize circular segments with our free, accurate online tool. Perfect for students, engineers, designers, and math enthusiasts.
Arc Length
Calculation Steps
Arc length formula: s = r × θ (θ in radians)
Angle conversion: 90° × (π/180) = 1.570796 radians
Calculation: 5 × 1.570796 = 7.853982
Arc Visualization
Angle: 90°
What is Arc Length?
Arc length is the distance along the curved line that makes up a segment of a circle’s circumference. It’s an important concept in geometry, trigonometry, and many practical applications such as engineering, architecture, and design.
When you have a circle and a central angle (an angle whose vertex is at the center of the circle), the arc length is the portion of the circle’s circumference that corresponds to that angle. The relationship between arc length, radius, and angle is fundamental to circular geometry.
Arc Length Formula
The formula for calculating arc length depends on whether the angle is measured in degrees or radians:
For angles in radians:
s = r × θ
Where:
- s = arc length
- r = radius of the circle
- θ = central angle in radians
For angles in degrees:
s = (θ/360) × 2πr
Or equivalently:
s = (πrθ)/180
Where:
- s = arc length
- r = radius of the circle
- θ = central angle in degrees
- π = mathematical constant pi (approximately 3.14159)
Practical Applications of Arc Length
Engineering & Construction
Arc length calculations are essential in designing curved structures, arches, bridges, and circular components in machinery.
Navigation
In navigation, arc length helps calculate distances along the Earth’s surface, which is approximately spherical.
Computer Graphics
Arc length is used in rendering curved shapes, animations, and designing user interface elements.
Manufacturing
In manufacturing, arc length calculations help determine material requirements for curved components.
Frequently Asked Questions
What’s the difference between arc length and chord length?
Arc length is the distance along the curved portion of a circle between two points, while chord length is the straight-line distance between those same two points.
Can arc length be greater than the circumference?
No, arc length is always a portion of the circumference, so it cannot exceed the total circumference of the circle.
How do I convert between degrees and radians?
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
What is sector area and how is it related to arc length?
Sector area is the area of the portion of a circle enclosed by two radii and their intercepted arc. While arc length measures the curved distance, sector area measures the space inside that curved segment.