Corner Radius Calculator

Precise Geometry & Design Tools

Calculate the radius of an arc based on its chord width and height.

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What is a Corner Radius Calculator?

A corner radius calculator is a specialized tool used to determine the radius of a circular arc, often referred to as a “rounded corner” or “fillet.” It works by taking two simple measurements: the chord width and the arc height. This is particularly useful in fields like graphic design, engineering, woodworking, and web development, where precise curvature is essential. Instead of guessing or performing complex manual geometry, this calculator provides instant, accurate results.

Anyone needing to create or measure rounded shapes can benefit from a corner radius calculator. For instance, a designer might use it to replicate a physical object’s curve in a digital design, or a web developer could use it to understand the geometry behind a CSS `border-radius`. This tool demystifies the process, making it accessible to anyone who can measure a straight line.

Corner Radius Formula and Explanation

The calculation is based on a geometric theorem involving segments of a circle. When you have a chord and the height (also known as the sagitta) of the corresponding arc, you can construct a right-angled triangle within the full circle. The corner radius calculator uses the following formula:

Radius (R) = (W² / 8H) + (H / 2)

This formula accurately computes the radius of the circle from which the arc is a segment. The derived values, like arc length and central angle, are then calculated using this radius. For those interested in the math, you can learn more by checking out a resource on how to calculate arc length.

Formula Variables

Description of variables used in the corner radius calculation.

Variable	Meaning	Unit (Auto-Inferred)	Typical Range
R	Radius	px, in, cm, mm	Positive number
W	Chord Width	px, in, cm, mm	Positive number
H	Arc Height (Sagitta)	px, in, cm, mm	Positive number, smaller than W
L	Arc Length	px, in, cm, mm	Positive number, greater than W
θ	Central Angle	Degrees	0° to 180°

Practical Examples

Example 1: Web Design Element

A UI designer needs to create a button with a specific rounded edge. They measure an existing design element and find the rounded part has a chord width of 80 pixels and an arc height of 10 pixels.

• Inputs: Chord Width (W) = 80 px, Arc Height (H) = 10 px
• Unit: Pixels (px)
• Results:
  • Radius (R): (80² / (8 * 10)) + (10 / 2) = (6400 / 80) + 5 = 80 + 5 = 85 px
  • This means they should use `border-radius: 85px;` in their CSS to get a perfect circular arc matching these dimensions. For more on this, our CSS border-radius guide is a great resource.

Example 2: Woodworking Project

A woodworker is building a custom shelf and wants to round the corners. They decide the rounded edge should span 6 inches across (chord width) and come in by 0.75 inches from the theoretical sharp corner (arc height).

• Inputs: Chord Width (W) = 6 in, Arc Height (H) = 0.75 in
• Unit: Inches (in)
• Results:
  • Radius (R): (6² / (8 * 0.75)) + (0.75 / 2) = (36 / 6) + 0.375 = 6 + 0.375 = 6.375 inches
  • The woodworker now knows they need to set their compass or router jig to a 6.375-inch radius to cut the perfect curve. This is a common application in CAD and engineering as well.

How to Use This Corner Radius Calculator

Using this tool is straightforward. Follow these steps for an accurate calculation:

1. Measure Chord Width (W): First, measure the straight-line distance spanning the start and end of the rounded corner. Enter this value into the “Chord Width” field.
2. Measure Arc Height (H): Next, find the midpoint of the chord line. Measure the distance from this midpoint to the peak of the curve. This is your arc height. Enter it into the “Arc Height” field. An accurate height is critical for a precise arc radius from chord and height calculation.
3. Select Units: Choose the unit of measurement (e.g., pixels, inches, cm) you used for your measurements from the dropdown menu. This ensures the output is correctly labeled.
4. Interpret the Results: The calculator will instantly display the corner radius, arc length, and the central angle of the curve. The primary result, the radius, is what you’ll typically need for design or fabrication.

Key Factors That Affect Corner Radius

• Chord Width: A wider chord, for the same arc height, results in a much larger radius. The radius increases with the square of the width.
• Arc Height: This has an inverse relationship with the radius. A smaller arc height (a flatter curve) for a given chord width results in a significantly larger radius.
• Ratio of Width to Height: The ratio between W and H is the most critical factor. A small H relative to W indicates a very large radius, while an H that is half of W (forming a semicircle) results in a radius equal to H.
• Measurement Accuracy: The formula is sensitive to small changes, especially in the arc height. A small measurement error in ‘H’ can lead to a large difference in the calculated radius, so precision is key.
• Units Consistency: Ensure both width and height are measured in the same units. Mixing units (e.g., inches for width and cm for height) without conversion will lead to incorrect results.
• Assumption of a Circular Arc: This corner radius calculator assumes the curve is a perfect segment of a circle. If the curve is elliptical or a more complex spline, the calculated radius will only be an approximation. Our SVG path tutorial explains more complex curves.

Frequently Asked Questions (FAQ)

1. What is the difference between radius and diameter?
The radius is the distance from the center of a circle to any point on its edge. The diameter is the distance across the circle passing through the center, which is always twice the radius.

2. Can I use this as a fillet radius formula calculator?
Yes, absolutely. In engineering and CAD, a rounded corner is often called a fillet. The geometry is the same, making this tool a perfect fillet radius formula calculator.

3. What if my corner isn’t a perfect circle?
This calculator assumes a constant radius (a circular arc). If your curve is an ellipse or a free-form spline, the result will be an approximation. You might need more advanced tools for non-circular curves.

4. Why does a small height give a huge radius?
As a curve gets flatter (smaller height for the same width), it more closely resembles a straight line. A straight line can be thought of as an arc of a circle with an infinite radius. Therefore, a very flat arc has a very large radius.

5. How does this differ from a CSS border-radius calculator?
A typical CSS `border-radius` property sets the radius directly. This tool, however, is a round corner calculator that works in reverse: it finds the radius from the physical shape of the curve, which is useful for replicating existing designs.

6. What is the ‘Central Angle’?
This is the angle formed at the center of the circle by the two radii that connect to the endpoints of the arc. It tells you how much of the circle the arc covers.

7. Can the arc height be larger than the chord width?
No. The maximum possible height for a given chord width is half the width (W/2). This occurs when the arc is a perfect semicircle, and the radius is equal to the height.

8. Is it possible to use this as an SVG arc calculator?
Yes. The calculated radius (R) can be used directly as the `rx` and `ry` values in an SVG path’s arc command (`A rx,ry …`). This tool helps you find the correct radius for your SVG drawings.