Length of Indicated Arc Calculator
Easily calculate the length of an arc of a circle with our free online Length of Indicated Arc Calculator. Input radius and central angle (degrees or radians).
Arc Length Calculator
Arc Length vs. Angle
Arc Length as Central Angle Changes (for Radius 10 and 15)
Example Arc Lengths
Arc Lengths for Different Radii and Angles (in Degrees)
| Radius (r) | Angle (θ) | Arc Length (s) |
|---|---|---|
| 5 | 30 | 2.62 |
| 5 | 90 | 7.85 |
| 5 | 180 | 15.71 |
| 10 | 30 | 5.24 |
| 10 | 90 | 15.71 |
| 10 | 180 | 31.42 |
| 15 | 30 | 7.85 |
| 15 | 90 | 23.56 |
| 15 | 180 | 47.12 |
What is a Length of Indicated Arc Calculator?
A Length of Indicated Arc Calculator is a tool used to determine the distance along the curved line forming an arc of a circle. An arc is a portion of the circumference of a circle. To find its length, you need to know the radius of the circle and the central angle that the arc subtends (the angle formed by two radii connecting the center to the arc's endpoints).
This calculator is useful for students, engineers, designers, and anyone working with circular shapes or paths. It simplifies the process of applying the arc length formula, especially when converting between degrees and radians. Our finding length of indicated arc calculator provides quick and accurate results.
Who Should Use It?
- Students: For geometry, trigonometry, and calculus homework and understanding.
- Engineers: In fields like mechanical, civil, and surveying for designing curved parts or paths.
- Designers: For graphic design, architecture, and manufacturing involving circular elements.
- Navigators: Although more complex calculations are often used, basic arc length concepts are relevant.
Common Misconceptions
A common misconception is confusing arc length with the chord length (the straight line distance between the endpoints of the arc). The arc length is always longer than the chord length unless the angle is zero. Another is incorrectly using the angle in degrees directly in the formula s = r * θ, which requires the angle to be in radians.
Length of Indicated Arc Calculator Formula and Mathematical Explanation
The length of an arc (s) of a circle is directly proportional to its central angle (θ) and the radius (r) of the circle.
The fundamental formula for arc length when the central angle θ is measured in radians is:
s = r * θ
Where:
sis the arc length.ris the radius of the circle.θis the central angle in radians.
If the central angle is given in degrees, it must first be converted to radians before using the formula. The conversion is:
Angle in radians = Angle in degrees * (π / 180)
So, if the angle is in degrees, the formula becomes:
s = r * (Angle in degrees * π / 180)
Our finding length of indicated arc calculator handles this conversion automatically based on your input.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| s | Arc Length | Length units (e.g., m, cm, inches) | 0 to 2πr (Circumference) |
| r | Radius | Length units (e.g., m, cm, inches) | Positive values |
| θ | Central Angle | Radians or Degrees | 0 to 2π radians or 0 to 360 degrees |
| π | Pi | Dimensionless constant | ~3.14159 |
Practical Examples (Real-World Use Cases)
Example 1: Designing a Curved Path
An architect is designing a curved garden path that is part of a circle with a radius of 20 meters. The path subtends a central angle of 60 degrees. What is the length of the path?
- Radius (r) = 20 m
- Angle (θ) = 60 degrees
First, convert the angle to radians: 60 * (π / 180) = π/3 radians ≈ 1.047 radians.
Arc Length (s) = 20 * (π/3) ≈ 20 * 1.047 ≈ 20.94 meters.
Using the finding length of indicated arc calculator with r=20 and angle=60 degrees gives s ≈ 20.94 m.
Example 2: Calculating Distance on a Circular Track
A runner runs along a circular track with a radius of 50 meters. If the runner covers a sector with a central angle of 2 radians, what distance did they run along the arc?
- Radius (r) = 50 m
- Angle (θ) = 2 radians
Arc Length (s) = 50 * 2 = 100 meters.
The runner covered 100 meters along the track. The Length of Indicated Arc Calculator quickly confirms this.
How to Use This Length of Indicated Arc Calculator
Using our finding length of indicated arc calculator is straightforward:
- Enter the Radius (r): Input the radius of the circle into the "Radius (r)" field.
- Enter the Central Angle (θ): Input the angle subtended by the arc into the "Central Angle (θ)" field.
- Select the Angle Unit: Choose whether the angle you entered is in "Degrees" or "Radians" from the dropdown menu.
- View Results: The calculator will automatically display the Arc Length (s), the angle in radians (if you entered degrees), the circle's circumference, and the area of the sector formed by the arc.
- Reset: Click the "Reset" button to clear the inputs and results to their default values.
- Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.
The results update in real-time as you change the input values, allowing for quick exploration of different scenarios.
Key Factors That Affect Length of Indicated Arc Results
Several factors directly influence the calculated arc length:
- Radius (r): The arc length is directly proportional to the radius. If you double the radius while keeping the angle constant, the arc length doubles.
- Central Angle (θ): The arc length is also directly proportional to the central angle. A larger angle (for the same radius) results in a longer arc.
- Angle Unit: It is crucial to specify the correct unit for the angle (degrees or radians). The formula s = r * θ only works directly if θ is in radians. Our Length of Indicated Arc Calculator handles the conversion if you input degrees.
- Measurement Accuracy: The accuracy of the arc length depends on the accuracy of the input radius and angle measurements.
- Value of Pi (π): The precision of the value of Pi used in the calculation (when converting from degrees) affects the final result, though most calculators use a sufficiently precise value.
- Context of the Problem: Understanding whether you need the arc length (curved distance) or chord length (straight distance) is important in practical applications.
Using a reliable finding length of indicated arc calculator ensures these factors are handled correctly.
Frequently Asked Questions (FAQ)
- What is an arc?
- An arc is a portion of the circumference of a circle.
- What is the formula for arc length in degrees?
- The formula for arc length when the angle is in degrees is s = r * (θ * π / 180), where s is arc length, r is radius, and θ is the angle in degrees.
- What is the formula for arc length in radians?
- When the angle is in radians, the formula is simpler: s = r * θ.
- How do I convert degrees to radians?
- To convert degrees to radians, multiply the degrees by π/180.
- How do I convert radians to degrees?
- To convert radians to degrees, multiply the radians by 180/π.
- Can the arc length be greater than the circumference?
- No, the arc length is a portion of the circumference, so it cannot be greater than the circumference (which corresponds to an angle of 360 degrees or 2π radians).
- What is the difference between arc length and chord length?
- Arc length is the distance along the curve of the arc, while chord length is the straight-line distance between the two endpoints of the arc.
- Does this finding length of indicated arc calculator also find the area of the sector?
- Yes, as an intermediate result, it calculates the area of the sector formed by the arc using the formula Area = 0.5 * r² * θ (with θ in radians).
Related Tools and Internal Resources
If you found the Length of Indicated Arc Calculator useful, you might also be interested in these related tools and resources:
- Area of Sector Calculator: Calculate the area of a sector of a circle based on radius and angle.
- Circumference Calculator: Find the circumference of a circle given its radius or diameter.
- Radians to Degrees Converter: Convert angles from radians to degrees and vice-versa.
- Circle Formulas: A summary of important formulas related to circles, including area, circumference, arc length, and sector area.
- Geometry Calculators: Explore a range of calculators for various geometric shapes and problems.
- Math Tools: A collection of mathematical calculators and converters.