Arc Length Calculator
Enter the radius and the central angle of a circle (in degrees, radians or gradians) and this calculator instantly returns the arc length using s = rθ, with the full calculation shown.
Arc Length from Radius and Central Angle
Arc Length Formula
Where:
- \(s\) — arc length (in the same unit as the radius)
- \(r\) — radius of the circle
- \(\theta\) — central angle in radians (the angle between the two radii that cut off the arc)
The formula says that an arc's length is directly proportional to both the radius and the angle: doubling either one doubles the arc length. If your angle is given in degrees or gradians, it must first be converted to radians, because \(1\) radian is defined as the angle that subtends an arc equal in length to the radius.
Arc Length – Calculation Example
Example: a circle has a radius of 5 m, and you need the length of the arc cut off by a central angle of 60°.
\[ \theta = 60 \cdot \frac{\pi}{180} = \frac{\pi}{3} \approx 1.0472 \text{ rad} \]
\[ s = r \cdot \theta = 5 \times 1.0472 = 5.236 \text{ m} \]
Result: the arc length is approximately 5.236 m (exactly \(\frac{5\pi}{3}\) m, about 17.5% of the full circumference \(2\pi r \approx 31.416\) m).
How to Use the Arc Length Calculator
- Enter the Radius (r) — the distance from the centre of the circle to the arc — and pick its unit (m, cm, mm, in or ft).
- Enter the Central angle (θ) between the two radii that cut off the arc, and choose its unit:
deg,radorgrad. - Read the Arc length in the results panel, expressed in the same unit as your radius.
- Check the Calculation line to see the angle converted to radians and the multiplication \(s = r\theta\) performed step by step.
What Is Arc Length?
An arc is a portion of a circle's circumference between two points. The arc length is the distance you would travel if you walked along the circle from one endpoint of the arc to the other — not the straight-line distance between them (that would be a chord).
The size of an arc is fixed by two quantities only: the radius of the circle and the central angle that subtends it. The central angle is the angle formed at the centre of the circle by the two radii drawn to the arc's endpoints. Because a full circle corresponds to \(2\pi\) radians, an arc covering the whole circle has length \(2\pi r\) — the familiar circumference formula — and any smaller angle takes a proportional slice of it.
This is why the angle must be in radians for \(s = r\theta\) to work. A radian is precisely the angle whose arc length equals the radius, so \(\theta\) radians literally means "\(\theta\) radius-lengths along the curve." In degrees, the same relationship requires an extra factor of \(\pi/180\), which this calculator handles automatically: enter the angle in any of the three supported units and it converts for you.
Arc length appears everywhere a curve is measured: the distance travelled along a circular track, the length of a bent pipe or sheet-metal bend, cable wound on a spool, or the length of a clock hand's tip path over a given time. It is also the foundation of curvilinear motion in physics and of line integrals in calculus.
Common Central Angles and Their Arcs
| Central angle | Fraction of circle | Arc length |
|---|---|---|
| 30° | 1/12 | \(\frac{\pi r}{6} \approx 0.524\,r\) |
| 45° | 1/8 | \(\frac{\pi r}{4} \approx 0.785\,r\) |
| 60° | 1/6 | \(\frac{\pi r}{3} \approx 1.047\,r\) |
| 90° | 1/4 | \(\frac{\pi r}{2} \approx 1.571\,r\) |
| 180° | 1/2 | \(\pi r \approx 3.142\,r\) |
| 360° | 1 | \(2\pi r \approx 6.283\,r\) |
Note that for angles up to 90°, the arc is only slightly longer than the radius — the geometry of radians keeps these numbers intuitive.
Assumptions, Limitations and Tips
- Circle only. The formula \(s = r\theta\) applies to circular arcs. For elliptical or arbitrary curves, arc length must be computed with calculus (integration), not with this tool.
- Angles larger than a full turn. If you enter more than 360° (or \(2\pi\) rad), the calculator still returns \(r\theta\); the arc simply wraps around the circle more than once — the calculation panel flags this for you.
- Same unit in and out. The result is always given in the unit you chose for the radius. Convert beforehand if you need the answer in a different unit.
- Sector vs. arc. This tool measures the curved edge only. If you need the area of the pie-slice region bounded by the arc and the two radii, use the sector area formula \(A = \tfrac{1}{2}r^2\theta\).
- Negative angles. A negative angle is treated as an orientation (clockwise), and the arc length comes out negative; enter the absolute value if you only care about the magnitude.
Related Concepts
Arc length is a classic example of direct variation: \(s\) varies directly with both \(r\) and \(\theta\), so the relationship forms a straight-line proportion. When working through the final multiplication \(r \times \theta_{\text{rad}}\) by hand, you can check it with the multiplication calculator, and the degree-to-radian conversion step is a simple division by \(180/\pi\).
Frequently Asked Questions
How do I calculate arc length by hand without a calculator?
Multiply the radius by the angle in radians: \( s = r\theta \). If your angle is in degrees, first convert it with \( \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180} \). For example, with \( r = 10 \) cm and \( \theta = 45° \): \( 45 \cdot \frac{\pi}{180} = \frac{\pi}{4} \approx 0.7854 \) rad, so \( s = 10 \times 0.7854 = 7.854 \) cm. Using \( \pi \approx 3.1416 \) is accurate enough for most hand calculations.
What is the difference between an arc and a chord?
An arc is the curved distance along the circle between two points, while a chord is the straight-line distance between them. For a central angle \( \theta \), the chord length is \( c = 2r\sin(\theta/2) \), which is always shorter than the arc \( s = r\theta \). For small angles they are nearly equal; at 90° with radius 1, the arc is 1.571 but the chord is only 1.414. This calculator returns the arc, not the chord.
Can I find the radius or the angle if I already know the arc length?
Yes, just rearrange \( s = r\theta \). The radius is \( r = s / \theta \) and the angle in radians is \( \theta = s / r \). To get the angle in degrees, multiply the radian result by \( 180/\pi \). For instance, if \( s = 12 \) cm and \( r = 9 \) cm, then \( \theta = 12/9 \approx 1.333 \) rad \( \approx 76.4° \).
What is the arc length of a semicircle or a quarter circle?
A semicircle (180°) has arc length \( \pi r \), exactly half the circumference \( 2\pi r \). A quarter circle (90°) has arc length \( \frac{\pi r}{2} \). So a semicircle with radius 4 m is \( 4\pi \approx 12.566 \) m long, and a quarter circle with the same radius is about 6.283 m. These fractions come straight from the proportion \( s = \frac{\theta}{360} \cdot 2\pi r \) when \( \theta \) is in degrees.
Can I enter the diameter instead of the radius?
Not directly — the formula \( s = r\theta \) needs the radius. If you know the diameter \( d \), halve it first: \( r = d/2 \), then compute \( s = \frac{d}{2}\theta \). For example, a 20 cm diameter with a 30° angle gives \( r = 10 \) cm and \( s = 10 \cdot \frac{\pi}{6} \approx 5.236 \) cm. Forgetting to halve the diameter is one of the most common mistakes and doubles the true answer.
Why is my answer wrong when I enter the angle in degrees?
The formula \( s = r\theta \) only works when \( \theta \) is in radians. If you multiply the radius by a degree value directly, the result is far too small: 60 degrees instead of \( \pi/3 \approx 1.047 \) radians understates the arc by a factor of about 57. Always check that the angle unit selector matches your input. If you compute by hand, convert degrees with \( \theta \cdot \pi/180 \) before multiplying.
How do gradians convert to radians?
A full turn equals 400 gradians, so one gradian is \( \frac{\pi}{200} \) radians, and the conversion is \( \theta_{\text{rad}} = \theta_{\text{grad}} \cdot \frac{\pi}{200} \). For example, 100 grad (a right angle) equals \( \pi/2 \approx 1.5708 \) rad, giving an arc of \( \frac{\pi r}{2} \). Gradians appear mainly in surveying and some European engineering practice; most everyday problems use degrees or radians.