Calculating Lenth Using Degrees
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Reviewed by Calculator Editorial Team
When working with angles and lengths in geometry, physics, or engineering, you may need to calculate the length of an arc or chord based on a given angle. This guide explains how to calculate length using degrees, provides a practical calculator, and offers real-world examples.
What is Degree Length?
Degree length refers to the measurement of an arc or chord in a circle where the angle is given in degrees. This concept is commonly used in:
- Geometry to calculate arc lengths and chord lengths
- Physics for circular motion and projectile paths
- Engineering in design and construction
- Navigation and surveying
The key difference between arc length and chord length is that an arc follows the curve of the circle, while a chord is the straight line connecting two points on the circumference.
Formula
Arc Length Formula
Arc length (L) = (θ/360) × 2πr
Where:
- θ = central angle in degrees
- r = radius of the circle
- π ≈ 3.14159
Chord Length Formula
Chord length (C) = 2r × sin(θ/2)
Where:
- θ = central angle in degrees
- r = radius of the circle
Assumptions
These formulas assume:
- The angle is measured in degrees
- The circle is perfect (no distortion)
- The radius is constant
How to Use the Calculator
- Enter the central angle in degrees (0-360)
- Enter the radius of the circle in your preferred units
- Select whether you want to calculate arc length or chord length
- Click "Calculate" to see the result
- Use the "Reset" button to clear all inputs
The calculator will display the calculated length in the same units as the radius you entered. The result card provides both the numerical value and a plain English explanation of what the number means.
Examples
Example 1: Arc Length Calculation
If you have a circle with radius 10 meters and a central angle of 90 degrees:
Arc length = (90/360) × 2π × 10 ≈ 15.708 meters
This means if you walk along the edge of a 90-degree arc in a 10-meter radius circle, you would cover approximately 15.7 meters.
Example 2: Chord Length Calculation
Using the same circle (radius 10 meters) with a 90-degree angle:
Chord length = 2 × 10 × sin(45) ≈ 14.142 meters
The straight-line distance between the two endpoints of the 90-degree arc is about 14.1 meters.
| Angle (degrees) | Arc Length (m) | Chord Length (m) |
|---|---|---|
| 30 | 2.618 | 5.236 |
| 60 | 5.236 | 10.000 |
| 90 | 7.854 | 14.142 |
| 120 | 10.472 | 17.321 |
FAQ
What's the difference between arc length and chord length?
Arc length follows the curve of the circle, while chord length is the straight-line distance between two points on the circumference. For small angles, the difference is negligible, but for larger angles, the arc length is always longer than the chord length.
Can I use these formulas for partial circles?
Yes, these formulas work for any angle between 0 and 360 degrees. For angles greater than 180 degrees, you'll be calculating the longer arc or chord.
What units should I use for the radius?
You can use any consistent unit (meters, inches, miles, etc.), but the result will be in the same units as the radius. For example, if you enter the radius in centimeters, the length will be in centimeters.