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Simply add the required values into this calculator to find the arc length of a curve, central angle, diameter, and more!
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When any two of the following values are known, you can use the arc length calculator to find the length of an arc and other related measurements:
Arc length is the distance along the curved line forming part of a circle between two points. Calculating the length of an irregular curve is called the rectification of a curve.
For a circle, if the central angle is \(\theta = 360^\circ\) or \(2\pi\) radians, the arc length equals the circumference:
\(C = 2\pi r\)
For a central angle \(\theta\) in radians, the proportional relationship is:
\(\frac{L}{\theta} = \frac{C}{2\pi} = r \quad \Rightarrow \quad L = r \cdot \theta\)
⚠️ Note: Ensure \(\theta\) is in radians. If the angle is in degrees, convert first:
\(\theta_\text{radians} = \theta_\text{degrees} \times \frac{\pi}{180}\)
There are two common methods:
You can also use an arc length calculator for quick results.
Find the arc length and sector area for a circle with:
Solution:
Step 1: Arc Length
\( s = r \cdot \theta = 50 \cdot \frac{\pi}{4} = \frac{50\pi}{4} = \frac{25\pi}{2} \approx 39.27 \, \text{cm} \)
Step 2: Sector Area
\( A = \frac{1}{2} r^2 \theta = \frac{1}{2} \cdot 50^2 \cdot \frac{\pi}{4} = \frac{1250 \pi}{4} \approx 981.75 \, \text{cm}^2 \)
If the sector area \(A\) and central angle \(\theta\) (in radians) are known:
\( L = \theta \cdot \sqrt{\frac{2A}{\theta}} \)
⚠️ Note: Convert \(\theta\) to radians if given in degrees.
Sector area: \( A = 300{,}000 \, \text{cm}^2 \), central angle: \( 60^\circ \). Find the arc length.
Solution:
Step 1: Convert area to square meters:
\( 300{,}000 \, \text{cm}^2 = 300{,}000 / 10{,}000 = 30 \, \text{m}^2 \)
Step 2: Convert angle to radians:
\( 60^\circ = 60 \cdot \frac{\pi}{180} = 1.0472 \, \text{radians} \)
Step 3: Apply the formula:
\( L = \theta \cdot \sqrt{\frac{2A}{\theta}} = 1.0472 \cdot \sqrt{\frac{2 \cdot 30}{1.0472}} \)
\( L = 1.0472 \cdot \sqrt{57.295} = 1.0472 \cdot 7.569 \approx 7.926 \, \text{m} \)
No, the angle measures the span between two radii, while the arc length is the distance along the circle between the two points.
The chord length is the straight line connecting two points on a circle, whereas the arc length is the curved distance along the circle between the same points.
Wikipedia: Arc Length
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