Mathematics, Statistics & Geometry

Arc Length of Curve Calculator

Calculate the arc length of a circular sector. Find the distance along the curved edge, sector area, and chord length with full step-by-step workings.

Arc Length
7.854
Sector Area39.27
Chord Length7.654
Angle in Radians0.785
Calculation StepsRadius (r) = 10 Angle (θ) = 45° = 45 * π / 180 = 0.785398 rad Arc Length (s) = r * θ s = 10 * 0.785398 = 7.853982 Area = 0.5 * r² * θ = 39.269908 Chord = 2r * sin(θ/2) = 7.653669

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Precise Circular Measurements

The Arc Length calculator allows you to solve for the curved distance along any circular path. Essential for architecture, mechanical engineering, and basic geometry, this tool handles conversions between degrees and radians automatically and provides the corresponding sector area and chord length.

s=rθ\begin{aligned} s = r\theta \end{aligned}

Where:
s=
The distance along the curved edge of the circle
r=
The distance from the center to the edge
θ\theta=
The angle subtended at the center (in radians)

Calculating Arc Length Step-by-Step

  1. Identify the Radius (r): The distance from the center of the circle to the arc.
  2. Determine the Angle (θ): The opening of the 'slice'.
  3. Convert to Radians: If your angle is in degrees, multiply by π/180.
  4. Multiply: Arc Length (s) = r × θ.

Beyond the Arc: Sector and Chord

  • Sector Area: The area of the 'pie slice'. Formula: A = 0.5 × r² × θ.
  • Chord Length: The straight-line distance between the two endpoints of the arc. Formula: c = 2r × sin(θ/2).

Practical Applications

  • Civil Engineering: Calculating the length of a curved road or a bridge arch.
  • Mechanical Design: Determining the length of a belt on a pulley system.
  • Navigation: Estimating travel distance along a great circle on the Earth's surface (spherical geometry).
  • Art and Design: Creating precise circular patterns and decorations.

Frequently Asked Questions

To find the arc length, multiply the radius of the circle by the central angle in radians. If your angle is in degrees, first convert it to radians by multiplying by π/180.

Arc length is the linear distance along the curve (the 'crust' of the pizza slice). Sector area is the two-dimensional space inside the slice (the 'cheese' and 'toppings').

While this calculator focuses on circular arcs (the most common requirement), the arc length of any smooth curve can be calculated using calculus by integrating the square root of 1 + (f'(x))² over the desired interval.

A radian is a unit of angular measure based on the radius of a circle. One radian is the angle created when the arc length equals the radius. There are exactly 2π radians in a full circle (360°).

The circumference of a circle is just a special case of arc length where the angle is a full 360° (2π radians). Thus, Circumference = r * 2π = 2πr.

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