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Arc Length Formula
Arc length is the distance along the curved edge of a circle between two points. It represents a fractional section of the total circumference. Calculating arc length is essential in geometry, trigonometry, design, and engineering. For an instant calculation, use the arc length calculator to solve for arc length, radius, or angle.
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What is the arc length formula?
The arc length formula determines the length of a circle's curved segment. The formula changes depending on whether your central angle is measured in radians or degrees.
When measured in radians, the formula is extremely simple because radians are naturally defined by the ratio of arc length to radius.
The arc length formulas
The two standard formulas for finding the arc length are:
Formula 1: Using Radians
s = rθ
Formula 2: Using Degrees
s = 2πr × (θ / 360)
The variables
- s = arc length
- r = radius of the circle
- θ = central angle (in radians for Formula 1, degrees for Formula 2)
- π = Pi (constant value ≈ 3.14159)
The degree version of the formula simply multiplies the full circumference (2πr) by the fraction of the circle that the angle covers (θ / 360).
Worked example: Calculating arc length
Let's calculate the arc length of a curved segment of a circle that has a radius of 6 inches and a central angle of 60 degrees.
- Identify the radius and angle: r = 6 in and θ = 60°.
- Choose the degree formula: s = 2 × π × r × (θ / 360).
- Substitute the values: s = 2 × π × 6 × (60 / 360).
- Simplify the angle fraction: 60 / 360 = 1/6.
- Multiply the values: s = 12π × (1/6) = 2π in.
- Calculate decimal value: s ≈ 2 × 3.14159 ≈ 6.28 in.
The arc length of the curved segment is exactly 2π inches, or approximately 6.28 inches.
Common arc length mistakes
- Using degrees in the radian formula. If you plug a degree value (e.g. 45) directly into s = rθ, your result will be 57 times larger than the correct answer. You must convert degrees to radians first, or use the degree-specific formula.
- Using diameter instead of radius. The radius is half of the diameter. If your problem provides a diameter, divide it by 2 before plugging it into the equation.
- Miscalculating the slice fraction. Ensure your division for θ / 360 is evaluated correctly before multiplying by the circumference.
Run the numbers
Explore circle and geometry calculators to verify your answers:
Frequently asked questions
An arc is a portion of the circumference (the outer boundary) of a circle. Its length is determined by the size of the circle (radius) and the angle forming the slice.
Degrees divide a circle into 360 equal parts. Radians measure angles based on the radius length. One full circle equals 360 degrees or 2π radians. The arc length formula is simpler when using radians.
Multiply the degree angle by π and divide by 180. For example, 90 degrees equals 90 × (π ÷ 180) = π/2 radians (approx. 1.57 radians).
No. Since an arc is a slice of a circle, the maximum arc length equals the full circumference when the angle is 360 degrees (or 2π radians).
The arc length calculator accepts radius and angle (in degrees or radians) and computes the exact and decimal arc lengths immediately, showing the formulas.