Scallop Height Calculator
Calculate the height of a circular segment (scallop) based on its width and the radius of the circle it belongs to. This tool is ideal for geometry, architecture, and CNC machining applications.
Scallop Height (h)
Arc Length
12.870
Segment Area
16.350
Central Angle
73.74°
Copied!
Visual Representation
What is a Scallop Height Calculator?
A scallop height calculator is a tool used to determine the height of a circular segment. In geometry, a circular segment is a region of a circle "cut off" from the rest of the circle by a chord. This shape is often referred to as a "scallop" in design, architecture, and particularly in CNC machining, where it describes the texture left on a surface by a round-nosed cutting tool. The height of this scallop (also known as the sagitta) is the maximum distance from the chord to the arc. This calculator helps you find this height (h) if you know the radius (R) of the circle and the width (c) of the scallop.
This calculation is essential for architects designing arched windows, engineers analyzing fluid in pipes, and machinists planning toolpaths to achieve a desired surface finish. Using a reliable scallop height calculator ensures precision and saves time on complex manual calculations.
Scallop Height Formula and Explanation
The calculation for scallop height is derived from the Pythagorean theorem. By creating a right-angled triangle with the radius (R) as the hypotenuse, half the chord length (c/2) as one leg, and the distance from the circle's center to the chord (R-h) as the other leg, we can solve for the height (h).
The formula is: h = R – √(R² – (c/2)²).
This formula directly computes the height of the circular segment. Our scallop height calculator uses this exact formula for instant and accurate results. For related calculations, check out our surface finish calculator.
Variables Table
| Variable | Meaning | Unit (Auto-Inferred) | Typical Range |
|---|---|---|---|
| h | Scallop Height (Sagitta) | Length (mm, in, etc.) | 0 to R |
| R | Circle Radius | Length (mm, in, etc.) | Greater than 0 |
| c | Scallop Width (Chord Length) | Length (mm, in, etc.) | 0 to 2R |
Practical Examples
Understanding the scallop height calculator is easier with real-world examples.
Example 1: Architectural Arch
An architect is designing a decorative arched window. The arch is a segment of a large circle with a radius of 6 feet. The width of the window opening (the chord) is 8 feet.
- Input (R): 6 ft
- Input (c): 8 ft
- Calculation: h = 6 – √(6² – (8/2)²) = 6 – √(36 – 16) = 6 – √20 ≈ 1.528 ft
- Result: The height of the arch at its center is approximately 1.53 feet.
Example 2: CNC Machining
A machinist is using a 10mm diameter ball-nose end mill (so the radius R is 5mm). The stepover, which is the distance between adjacent passes (equivalent to the scallop width c), is set to 1mm to achieve a fine finish. What is the theoretical height of the scallop ridges left on the surface?
- Input (R): 5 mm
- Input (c): 1 mm
- Calculation: h = 5 – √(5² – (1/2)²) = 5 – √(25 – 0.25) = 5 – √24.75 ≈ 0.025 mm
- Result: The scallop height is 0.025 mm (or 25 microns), indicating a very smooth surface finish. For more details on machining parameters, you might find our milling speed and feed calculator useful.
How to Use This Scallop Height Calculator
Using our tool is straightforward. Follow these steps for an accurate calculation:
- Enter Circle Radius (R): Input the radius of the circle that the scallop is a part of.
- Enter Scallop Width (c): Input the chord length, which is the straight-line width of the scallop.
- Select Units: Choose the unit of measurement (e.g., inches, mm). Ensure that both your radius and width values are in this same unit.
- Interpret Results: The calculator instantly provides the primary result (Scallop Height, h) and several intermediate values like Arc Length and Segment Area. The visual chart will also update to reflect your inputs.
- Handle Errors: If you enter a width (c) that is greater than twice the radius (2R), the calculator will show an error, as this geometry is impossible.
Key Factors That Affect Scallop Height
Several factors influence the calculated scallop height. Understanding them helps in both design and practical applications.
- Circle Radius (R): For a fixed scallop width, a larger radius results in a smaller scallop height. The curve is flatter over the same distance.
- Scallop Width (c): For a fixed radius, a larger scallop width results in a significantly larger scallop height. The scallop cuts deeper into the circle.
- Ratio of Width to Radius: The most critical factor is the c/R ratio. As this ratio approaches 2 (i.e., the width approaches the circle's diameter), the height approaches the radius.
- Unit Consistency: Mixing units (e.g., a radius in inches and a width in cm) will lead to incorrect results. Our scallop height calculator requires consistent units.
- Measurement Accuracy: The precision of your input values for radius and width directly impacts the accuracy of the output. Small errors in measurement can be magnified in the calculation.
- Tool Shape (in Machining): In CNC applications, the scallop height is directly related to the tool's radius (for a ball-nose tool) and the step-over distance. This is a primary concern for achieving a good surface finish.
Frequently Asked Questions (FAQ)
- What is another name for scallop height?
- Scallop height is also known as the "sagitta" of a circular segment. In machining, it's sometimes called "cusp height".
- What happens if the width is greater than twice the radius?
- A chord's length can never exceed the circle's diameter (2 x Radius). If you enter a width greater than 2R, the calculation is geometrically impossible, and our calculator will display an error message.
- How does scallop height relate to surface finish in CNC milling?
- In milling, scallop height is the measure of the ridges left between tool passes. A smaller scallop height means a smoother surface. Machinists use this calculation to determine the optimal "step-over" to achieve a desired finish quality. You can explore this further with a CNC machining calculator.
- Can I calculate the radius if I know the height and width?
- Yes, the formula can be rearranged to solve for the radius: R = (h² + (c/2)²) / (2h). This is useful in reverse-engineering scenarios.
- Does this calculator work for a semicircle?
- Yes. A semicircle is a special case where the width (c) is equal to the diameter (2R). If you input c = 2R, the calculator will correctly show the height (h) as being equal to the radius (R).
- What are the intermediate values shown in the results?
- Besides the height, our calculator shows the Arc Length (the distance along the curved edge), the Segment Area (the 2D area of the scallop), and the Central Angle (the angle formed at the center of the circle by the segment's endpoints).
- Is the visual chart drawn to scale?
- No, the chart is a representative diagram to help you visualize the relationship between R, c, and h. It adjusts its general shape based on your inputs but is not a precise scale drawing.
- How can I ensure my calculation is accurate?
- Double-check your input values for radius and width, and make sure they are both in the same unit of measurement. Our calculator uses a precise mathematical formula, so the accuracy of the output depends entirely on the accuracy of your input.
Related Tools and Internal Resources
If you found the scallop height calculator useful, you might also be interested in these related tools:
- Surface Finish Calculator: Specifically for CNC machining applications to determine surface roughness based on tool size and step-over.
- Circle Calculator: A comprehensive tool for calculating area, circumference, diameter, and other properties of a circle.
- Milling Speed and Feed Calculator: Helps determine the optimal settings for your milling machine operations.
- CNC Machining Calculator: A general-purpose calculator for various CNC-related tasks.