How To use the Arc Length Calculator,
Central Angle:
- Enter the value of the central angle in the “Central Angle” input field.
- Choose the unit for the central angle from the dropdown menu (“Degrees” or “Radians”).
Radius:
- Enter the value of the radius in the “Radius” input field.
- Choose the unit for the radius from the dropdown menu (“Millimeters,” “Centimeters,” “Inches,” “Feet,” or “Meters”).
Calculate:
- Click the “Calculate Arc Length” button.
Results:
- The calculator will display the arc length in millimeters, centimeters, inches, feet, and meters.
Make sure to enter valid numeric values greater than zero for both the central angle and the radius. The results will automatically update when you click the “Calculate Arc Length” button. If there are any issues or invalid inputs, an alert will inform you to correct the entries.
Formula to calculate arc length and some examples
The formula to calculate arc length (\(L\)) given the central angle (\(\theta\)) in radians and the radius (\(r\)) is:
\[ L = \theta \times r \]Find the arc length (\(L\)) for a circle with a central angle (\(\theta\)) of \( \frac{\pi}{3} \) radians and a radius (\(r\)) of 10 millimeters.
Answer:
Using the arc length formula:
\[ L = \frac{\pi}{3} \times 10 = \frac{10\pi}{3} \text{ millimeters} \]Therefore, the arc length is \( \frac{10\pi}{3} \) millimeters.
Calculate the arc length (\(L\)) for a circle with a central angle (\(\theta\)) of \(60\) degrees and a radius (\(r\)) of \(5\) centimeters.
Answer:
First, convert the central angle to radians (\(\theta\)): \(60\) degrees \(= \frac{\pi}{180} \times 60 = \frac{\pi}{3}\) radians.
Using the arc length formula:
\[ L = \frac{\pi}{3} \times 5 = \frac{5\pi}{3} \text{ centimeters} \]Therefore, the arc length is \( \frac{5\pi}{3} \) centimeters.
Derivation of arc length formula
Consider a circle with radius \(r\) and center at the origin. For a small portion of the circle subtended by a central angle \(\Delta \theta\), the arc length is given by:
\[ \Delta L = r \cdot \Delta \theta \]To find the arc length \(L\) for the entire circle, integrate \(\Delta L\) with respect to \(\theta\) over the interval \([0, \theta]\):
\[ L = \int_{0}^{\theta} r \, d\theta \]Evaluating the integral gives the formula for arc length:
\[ L = r \theta \]what is arc length?
Arc length refers to the distance along a curve or the curved path between two points on a curve. In the context of a circle or a portion of a circle, arc length is the measure of the distance along the circumference of the circle between two points, measured along the curved line.
For a complete circle, the arc length is equal to the circumference of the circle. The formula for calculating arc length () is given by:
where:
- is the arc length,
- is the central angle (in radians) subtended by the arc,
- is the radius of the circle.
In other words, the arc length is the product of the central angle and the radius. This formula can be used to find the length of any arc on a circle, given the measure of the central angle and the radius.