Find Radius from Chord Length Calculator
Enter the chord length and the sagitta (height of the circular segment) to calculate the radius of the circle.
What is the Find Radius from Chord Length Calculator?
A find radius from chord length calculator is a specialized tool used to determine the radius of a circle when you know the length of a chord and the sagitta (also known as the height of the circular segment formed by the chord). This is particularly useful in geometry, engineering, construction, and design where you might have measurements of a part of a circle but need to find the radius of the full circle it belongs to.
Anyone working with circular shapes or segments, such as architects, engineers, carpenters, machinists, and designers, can benefit from using a find radius from chord length calculator. It allows for quick and accurate radius determination without needing to measure the center of the circle, which might be impractical or impossible.
A common misconception is that you need the center of the circle or the full circumference to find the radius. However, with just the chord length and the sagitta, the radius can be precisely calculated using geometric formulas. Our find radius from chord length calculator simplifies this process.
Find Radius from Chord Length Calculator Formula and Mathematical Explanation
The relationship between the radius (R), the chord length (c), and the sagitta (h) of a circular segment is derived from the Pythagorean theorem applied to a right triangle formed by the radius, half the chord, and the distance from the center to the chord (R-h).
Consider a circle with radius R. A chord of length c is drawn, and the sagitta (height from the chord to the arc) is h. The distance from the center of the circle to the chord is R – h. Half the chord length is c/2. We can form a right-angled triangle with sides R (hypotenuse), c/2, and R – h.
By the Pythagorean theorem:
(R – h)² + (c/2)² = R²
R² – 2Rh + h² + c²/4 = R²
Subtracting R² from both sides:
-2Rh + h² + c²/4 = 0
Rearranging to solve for R:
2Rh = h² + c²/4
R = (h² / 2h) + (c²/4 / 2h)
R = h/2 + c²/8h
This is the formula used by the find radius from chord length calculator.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| R | Radius of the circle | Length (e.g., m, cm, in) | Positive values |
| c | Chord Length | Length (e.g., m, cm, in) | Positive values |
| h | Sagitta (Height of the segment) | Length (e.g., m, cm, in) | Positive values, typically less than R and often less than c/2 |
Practical Examples (Real-World Use Cases)
Example 1: Broken Circular Table
Imagine you have a fragment of a circular glass tabletop. You can measure the straight edge of the fragment (the chord, c) as 40 cm, and the greatest perpendicular distance from the chord to the curved edge (the sagitta, h) as 5 cm. To order a replacement top, you need the radius.
- Chord Length (c) = 40 cm
- Sagitta (h) = 5 cm
Using the find radius from chord length calculator or the formula R = (c² / 8h) + h/2:
R = (40² / (8 * 5)) + (5 / 2) = (1600 / 40) + 2.5 = 40 + 2.5 = 42.5 cm
The radius of the original table was 42.5 cm.
Example 2: Curved Archway
An architect is designing a curved archway with a span (chord length, c) of 6 meters and a height at the center (sagitta, h) of 1 meter. They need to determine the radius of the circle that forms this arc.
- Chord Length (c) = 6 m
- Sagitta (h) = 1 m
Using the find radius from chord length calculator:
R = (6² / (8 * 1)) + (1 / 2) = (36 / 8) + 0.5 = 4.5 + 0.5 = 5 meters
The radius of the arc is 5 meters. You can find more about arc calculations with our arc radius calculator.
How to Use This Find Radius from Chord Length Calculator
- Enter Chord Length (c): Input the measured length of the chord into the "Chord Length (c)" field.
- Enter Sagitta (h): Input the measured height from the midpoint of the chord to the arc into the "Sagitta (h) / Height" field.
- Calculate: The calculator will automatically update the results as you type or you can click the "Calculate" button.
- View Results: The calculated Radius (R) will be displayed prominently, along with intermediate values like c² / 8h and h / 2.
- Use the Chart: The chart dynamically shows how the radius changes with variations in chord length or sagitta around the entered values, helping you visualize the relationship. Check out our geometry tools for more visualizations.
- Reset or Copy: Use the "Reset" button to clear inputs to default values or "Copy Results" to copy the main result and inputs.
The find radius from chord length calculator gives you the radius based on the geometric relationship between these measurements.
Key Factors That Affect Find Radius from Chord Length Calculator Results
- Accuracy of Chord Length Measurement: A small error in measuring the chord length can lead to a noticeable difference in the calculated radius, especially if the sagitta is very small compared to the chord.
- Accuracy of Sagitta Measurement: The sagitta is often smaller than the chord length, and relative errors in its measurement can significantly impact the radius calculation, particularly the c²/8h term.
- Ratio of Sagitta to Chord Length: When the sagitta is very small relative to the chord, the radius becomes very large, and the calculation is more sensitive to small changes in h. Conversely, as h approaches R (for a semicircle, h=R, c=2R, but the formula is for h
- Units of Measurement: Ensure both chord length and sagitta are measured in the same units. The resulting radius will be in those same units.
- Assumption of a Perfect Circle: The formula assumes the segment is part of a perfect circle. If the curve is not perfectly circular, the calculated radius is an approximation.
- Measurement Points: Ensuring the sagitta is measured exactly from the midpoint of the chord and perpendicularly to the highest point of the arc is crucial for accuracy.
Frequently Asked Questions (FAQ)
- What is a chord of a circle?
- A chord is a straight line segment whose endpoints both lie on the circumference of the circle.
- What is the sagitta?
- The sagitta is the distance from the center of a chord to the arc of the circle, measured perpendicularly to the chord. It's the height of the circular segment.
- Can I use this calculator if I have the arc length instead of chord length?
- No, this specific find radius from chord length calculator requires the chord length and sagitta. If you have the arc length and chord length or sagitta, different formulas or iterative methods are needed, which you might find in our arc calculator.
- What if the sagitta is very small?
- If the sagitta is very small compared to the chord, the radius will be very large, indicating a very gentle curve. The calculation becomes more sensitive to errors in 'h'.
- What if the sagitta is equal to the radius?
- If the sagitta equals the radius, the chord length must be equal to the diameter (2R), and the segment is a semicircle. However, the formula R = h/2 + c²/8h is typically used when h < R.
- Can the sagitta be larger than the radius?
- No, the sagitta, as the height of the segment from the chord, cannot be larger than the radius for a single chord forming a segment less than a semicircle or a semicircle. If you are measuring from the chord to the 'far' side of the circle through the center, that's different.
- How accurate is the find radius from chord length calculator?
- The calculator is as accurate as the input measurements and the formula. The mathematical formula is exact for a perfect circle.
- Are there other ways to find the radius from a segment?
- Yes, if you know the chord length and the angle subtended by the chord at the center, or the arc length and the angle, you can also find the radius. See more at chord properties.