Arc Length Calculator

Understanding Arc Length

Arc length is the distance along a curved line, specifically a portion of the circumference of a circle.

The Arc Length Formula

For a circle with radius r and central angle θ (in radians):

$$s = r \times \theta$$

Where:

• s = arc length
• r = radius
• θ = central angle in radians

Converting Degrees to Radians

Since most people think in degrees, convert first:

$$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$$

Or use the combined formula:

$$s = r \times \theta_{deg} \times \frac{\pi}{180}$$

Rearranging the Formula

To find radius

$$r = \frac{s}{\theta}$$

To find angle

$$\theta = \frac{s}{r}$$

Related Calculations

Sector area

The area of a "pie slice" of a circle:

$$A = \frac{1}{2} r^2 \theta = \frac{1}{2} r \times s$$

Chord length

The straight-line distance between arc endpoints:

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

Common Arc Lengths

Angle	Arc Length
360° (full circle)	2πr
180° (semicircle)	πr
90° (quarter circle)	πr/2
60°	πr/3
45°	πr/4

Examples

Quarter circle

For r = 10 and θ = 90°:

$$s = 10 \times 90 \times \frac{\pi}{180} = 10 \times \frac{\pi}{2} = 5\pi \approx 15.71$$

Full circle

For r = 5 and θ = 360°:

$$s = 5 \times 2\pi = 10\pi \approx 31.42$$

This equals the circumference.

Applications

Architecture

• Designing curved walls and arches
• Calculating material for curved surfaces
• Staircase handrails

Engineering

• Road and railway curves
• Gear teeth design
• Belt and pulley systems

Navigation

• Great circle distances (spherical arcs)
• Flight path calculations
• Maritime routes

Sports

• Running track lane lengths
• Curved playing fields
• Trajectory analysis

Arc Length on a Sphere

For a sphere with radius R, the arc length between two points is:

$$s = R \times \theta$$

Where θ is the central angle in radians.

Parametric Arc Length

For a curve defined by x(t) and y(t):

$$s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$

Relationship to Circumference

The arc length as a fraction of circumference:

$$\frac{s}{2\pi r} = \frac{\theta}{2\pi}$$

Or in degrees:

$$\frac{s}{2\pi r} = \frac{\theta_{deg}}{360}$$

Tips for Calculations

1. Check units: Ensure angle is in radians for the basic formula
2. Verify reasonableness: Arc length should be less than circumference
3. Full circle check: If θ = 2π, s should equal 2πr