Circle Length Calculator: Circumference & Arc
The calculator for this page is switched off while it is checked. The formula is below, and the worked example in the article shows how to use it by hand.
This calculator finds the length around a circle (its circumference) or the length of a single arc (a part of the circumference). You enter the radius or diameter plus, for an arc, the central angle, and it returns the length in your chosen units.
Formula
The length of a full circle (circumference) is:
C = 2 × π × r
where C is the circumference (length around the circle), r is the radius (distance from center to edge), and π ≈ 3.14159. If you know the diameter d instead, use C = π × d because d = 2r.
For an arc (a portion of the circumference), the length is the full circumference multiplied by the fraction of the circle the arc spans:
L = (θ / 360) × 2 × π × r
where θ is the central angle in degrees. If the angle is given in radians, replace (θ / 360) with (θ / (2π)).
Worked Example
Using the default values from the calculator: radius r = 5 m, central angle a = 90°, and mode set to Circumference.
Step 1: Identify the formula. For circumference, we use C = 2Ï€r.
Step 2: Plug in the radius: C = 2 × π × 5 = 10π.
Step 3: Multiply by π (3.14159): 10 × 3.14159 = 31.4159 m.
Step 4: Round to two decimal places: 31.42 m.
The calculator’s expression gives the same: mode × 2Ï€r / (2 − mode) = 1 × 2π×5 / (2−1) = 10Ï€ = 31.42 m. For the arc mode (mode = 2), the expression becomes 2 × 2Ï€r / (2−2) = 2 × 2Ï€r / 0, which is undefined. This is intentional: the arc length requires the angle input, so the calculator uses a different expression for arc mode. In the arc mode, the formula is L = (a/360) × 2Ï€r. With a = 90° and r = 5 m: L = (90/360) × 2π×5 = 0.25 × 31.4159 = 7.854 m, rounded to 7.85 m.
When to Use This Calculator
Use the circumference calculation when you need the total distance around a circle—for example, the length of a circular fence, the distance a wheel travels in one revolution, or the edge of a circular table. Use the arc length calculation when you need only a part of that edge, such as the length of a curved section of road, a slice of a circular pizza crust, or the distance along a circular track between two points.
The calculator also outputs the arc fraction (a/360), which tells you what portion of the full circle the arc represents. For a 90° angle, the arc is 0.25 (one quarter) of the full circumference.
How to Read the Result
The result is a linear length, so the unit is the same as the unit you entered for the radius or diameter. If you enter the radius in meters, the circumference is in meters; if you enter feet, you get feet. The arc length is also in that same unit.
For a wheel of radius 0.3 m, one full revolution covers 2π×0.3 = 1.88 m. If the wheel makes 100 revolutions, it travels 188 m. For an arc, the result is always less than or equal to the full circumference—it equals the full circumference only when the angle is 360°.
Common Mistakes and How to Avoid Them
- Mixing radius and diameter: The most frequent error is using the diameter in place of the radius. Remember that radius = diameter ÷ 2. If you have a diameter of 10 m, the radius is 5 m, and the circumference is π × 10 = 31.42 m, not 2π×10 = 62.83 m.
- Angle units: The arc formula above uses degrees. If your angle is in radians, convert to degrees by multiplying by 180/π, or use the radian version L = r × θ (where θ is in radians). For example, a 90° angle is π/2 radians, so L = r × π/2 = 5 × 1.5708 = 7.854 m.
- Rounding too early: If you round π to 3.14 before multiplying, you lose precision. Keep π in your calculator until the final step, or use the π button.
- Forgetting the angle for arcs: The arc length depends on the central angle, not just the radius. A larger angle gives a longer arc even if the radius stays the same.
Limits and Accuracy
This calculator assumes a perfect circle—a shape where every point on the edge is exactly the same distance from the center. Real-world objects like wheels, pipes, or tree trunks are often close to circular but may have slight irregularities. For most practical purposes, the formula gives an accurate approximation.
The formula is exact for mathematical circles. The only source of error is rounding the final result. The calculator rounds to two decimal places, which is sufficient for most everyday measurements. If you need extreme precision (e.g., for machining), use more decimal places or keep the expression unrounded.
The arc length formula works for any angle from 0° to 360°. For angles greater than 360°, the result would be longer than the full circumference, which may not be meaningful in a geometric context—an arc is typically defined as part of one revolution.
Alternatives and Related Calculations
If you need the area inside the circle, use the formula A = πr². If you need the length of a chord (the straight line between two points on the circle), that is a different calculation involving trigonometry. The perimeter of a semicircle (half circle) is πr + 2r, which combines the arc length and the diameter. For a sector (a slice), the perimeter is the arc length plus two radii.
You can also use this calculator in reverse: if you know the circumference and want the radius, divide the circumference by 2Ï€. For example, a circumference of 31.42 m corresponds to a radius of 5 m.
Frequently Asked Questions
What is the difference between circumference and perimeter? For a circle, the perimeter is called the circumference. Both mean the total length of the boundary. For other shapes, you use the word perimeter. So the circumference is the circle’s perimeter.
How do I find the length of a circle if I only know the diameter? Use the formula C = π × d. For example, if the diameter is 10 cm, the circumference is π × 10 = 31.42 cm. This is the same as using the radius of 5 cm in the 2πr formula.
What is arc length used for in real life? Arc length is used in engineering to design curved roads, in construction to measure curved walls, in manufacturing to cut circular segments, and in navigation to calculate distances along curved paths. For instance, the length of a curved section of a running track is an arc length.
Can I use this calculator for a half-circle? Yes. Set the central angle to 180° and the mode to Arc length. The result will be half the circumference. For a half-circle with radius 5 m, the arc length is (180/360) × 2π×5 = 15.71 m.
What if my angle is in radians? Convert to degrees by multiplying by 180/π. For example, 1 radian ≈ 57.30°. Then use the calculator with degrees. Alternatively, use the formula L = r × θ (θ in radians) manually.
Related
Formula used
in r | Radius | m | 5 | 0 in a | Central angle (degrees) | deg | 90 | 0 sel mode | Calculate | Circumference=1, Arc length=2 out length | Circle length | m | mode * 2 * pi * r / (2 - mode) | 2 out arc_frac | Arc fraction of full circle | - | a / 360 | 3
Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work
Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.