Find Radius Of Arc Calculator

Easily calculate the radius of a circular arc using the chord length and the height (sagitta) of the arc. Enter the values below to get the radius.

Visual representation of the arc, chord, and height.

Input	Value	Calculated Radius
Chord Length	10	7.25
Arc Height	2

Table summarizing inputs and the calculated radius.

What is a Radius of Arc Calculator?

A Radius of Arc Calculator is a tool used to determine the radius of a circle of which a given arc is a part. It typically uses the chord length of the arc and the height (or sagitta) of the arc as inputs. Knowing the radius is crucial in various fields like engineering, architecture, optics, and even woodworking, where circular arcs are frequently encountered.

Anyone working with curved shapes that form part of a circle can benefit from a Radius of Arc Calculator. This includes engineers designing curved beams or roads, architects designing arches or domes, machinists shaping parts, or carpenters building curved furniture.

A common misconception is that you need the angle subtended by the arc to find the radius. While the angle can be used, the Radius of Arc Calculator often relies on more easily measurable linear dimensions: the chord length and the arc height.

Radius of Arc Calculator Formula and Mathematical Explanation

The most common formula used by a Radius of Arc Calculator when the chord length (c) and arc height/sagitta (h) are known is derived from the properties of a circle and the Pythagorean theorem.

Consider a circle with radius R. An arc is defined by a chord of length c and a height h (the sagitta). The height h is the distance from the midpoint of the chord to the arc itself, along a line passing through the center of the circle.

If we draw a radius from the center of the circle to one endpoint of the chord, and another line from the center perpendicular to the chord (passing through the midpoint of the chord and the arc), we form a right-angled triangle. The sides of this triangle are:

• The hypotenuse is the radius R.
• One leg is half the chord length (c/2).
• The other leg is the distance from the center to the chord, which is (R – h).

Using the Pythagorean theorem (a² + b² = c²):

(c/2)² + (R – h)² = R²

c²/4 + R² – 2Rh + h² = R²

c²/4 + h² = 2Rh

Solving for R:

R = (c²/4 + h²) / (2h)

R = c²/(8h) + h²/(2h)

R = c²/(8h) + h/2

Variables Used in the Radius of Arc Calculation

Variable	Meaning	Unit	Typical Range
R	Radius of the circle	Length (e.g., m, cm, inches)	Positive
c	Chord length	Length (e.g., m, cm, inches)	Positive
h	Arc height (sagitta)	Length (e.g., m, cm, inches)	Positive, h < R

Practical Examples (Real-World Use Cases)

Let’s see how the Radius of Arc Calculator works with some examples:

Example 1: Architectural Arch

An architect is designing a shallow arched window. The base of the arch (chord length) is 120 cm, and the height of the arch from the base is 10 cm.

• Chord Length (c) = 120 cm
• Arc Height (h) = 10 cm

Using the formula R = c²/(8h) + h/2:

R = (120²)/(8 * 10) + 10/2 = 14400/80 + 5 = 180 + 5 = 185 cm.

The radius of the circle forming the arch is 185 cm. Our Radius of Arc Calculator would provide this result.

Example 2: Curved Road Section

A civil engineer is measuring a curved section of a road. They measure a straight line (chord) between two points on the curve as 50 meters, and the perpendicular distance from the midpoint of this chord to the road curve is 1.5 meters.

• Chord Length (c) = 50 m
• Arc Height (h) = 1.5 m

R = (50²)/(8 * 1.5) + 1.5/2 = 2500/12 + 0.75 ≈ 208.33 + 0.75 = 209.08 meters.

The radius of the curve is approximately 209.08 meters. The Radius of Arc Calculator helps in quickly determining this.

How to Use This Radius of Arc Calculator

1. Enter Chord Length (c): Input the length of the straight line connecting the two endpoints of the arc into the “Chord Length (c)” field.
2. Enter Arc Height (h): Input the perpendicular distance from the midpoint of the chord to the highest point of the arc into the “Arc Height / Sagitta (h)” field.
3. View Results: The calculator will automatically update and display the “Radius (R)”, “Half Chord (c/2)”, and the ratio h/c in real-time.
4. Check Formula: The formula used for the calculation is displayed below the results.
5. Visualize: The chart below the results provides a visual representation of the arc, chord, and height within the context of the circle segment.
6. Reset: Use the “Reset” button to clear the inputs and results to their default values.
7. Copy: Use the “Copy Results” button to copy the input values and calculated results to your clipboard.

The results from the Radius of Arc Calculator can inform decisions about the curvature needed for a design, the material required, or the fit of a curved object.

Key Factors That Affect Radius of Arc Results

The calculated radius of an arc is directly influenced by the measurements of the chord length and arc height. Here are key factors:

1. Accuracy of Chord Length Measurement: A small error in measuring the chord length can lead to a significant difference in the calculated radius, especially for very flat arcs (small h).
2. Accuracy of Arc Height Measurement: The arc height is often smaller than the chord length, and errors in its measurement are magnified in the radius calculation (as h is in the denominator of the c²/(8h) term).
3. Ratio of Height to Chord: For very small h compared to c (very flat arcs), the radius becomes very large, and the calculation is more sensitive to small changes in h. Conversely, as h approaches R (a semicircle), the sensitivity changes.
4. Units of Measurement: Ensure both chord length and arc height are measured in the same units. The radius will be in the same unit.
5. Point of Height Measurement: The arc height must be measured precisely from the midpoint of the chord and perpendicular to it.
6. Nature of the Arc: The formula assumes the arc is part of a perfect circle. If the curve is elliptical or irregular, this formula will only provide an approximation of the local radius of curvature.

Understanding these factors is crucial for obtaining a reliable result from the Radius of Arc Calculator.

Frequently Asked Questions (FAQ)

Q1: What is the difference between arc height and sagitta?
A1: Arc height and sagitta refer to the same thing: the perpendicular distance from the midpoint of the chord to the arc. The Radius of Arc Calculator uses this value.

Q2: Can I use this calculator if I know the arc length and chord length instead of height?
A2: This specific Radius of Arc Calculator uses chord length and height. If you know the arc length and chord length, a different set of formulas (often involving solving transcendental equations or approximations) is needed, or you might use an Arc Length Calculator in reverse if it allows.

Q3: What if the arc is more than a semicircle (h > c/2)?
A3: The formula still works, but h would be the height from the chord to the far side of the circle, making it greater than the radius. Usually, when we talk about arc height, it’s for segments less than a semicircle (h < R, and for a segment h <= R). If h > R, it means the center is between the chord and the arc measured. The calculator is primarily for h < R.

Q4: Why does the radius become very large when the height is very small?
A4: A very small height for a given chord means the arc is very flat, indicating it’s part of a very large circle, hence a large radius. Our Radius of Arc Calculator shows this relationship.

Q5: Can I calculate the radius from the angle and chord length?
A5: Yes, if you know the angle (θ) subtended by the arc at the center, the radius R can be found using R = (c/2) / sin(θ/2). This Radius of Arc Calculator doesn’t use the angle directly.

Q6: How accurate is this Radius of Arc Calculator?
A6: The calculator performs the mathematical calculation accurately based on the formula. The accuracy of the result depends entirely on the accuracy of your input measurements for chord length and arc height.

Q7: What are the limitations of this calculator?
A7: It assumes the arc is a segment of a perfect circle and requires accurate chord and height measurements. It does not work directly with arc length or angle as primary inputs.

Q8: In what units should I enter the values?
A8: You can use any consistent unit of length (meters, centimeters, inches, feet, etc.) for both chord length and arc height. The calculated radius will be in the same unit. The Radius of Arc Calculator is unit-agnostic.