Corner Radius Calculator

A corner radius calculator is an essential tool for designers, engineers, and developers. This professional date-related web developer and SEO content strategist’s tool helps you determine the precise radius of a circular arc based on its chord length and sagitta (height), ensuring your designs are mathematically accurate and visually appealing. Use this corner radius calculator to achieve perfect curves in your projects.

Corner Radius (r)

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Formula Used: r = (s² + (c/2)²) / (2s)

Results Breakdown

Metric	Value	Description

This table provides a detailed breakdown of the inputs and calculated geometric properties of the corner radius.

What is a Corner Radius Calculator?

A corner radius calculator is a specialized tool used to determine the radius of a circle of which a given circular arc is a part. Instead of guessing or relying on visual estimation, this calculator uses geometric inputs—specifically the chord length and the sagitta (arc height)—to compute the exact radius. This ensures precision in various fields.

Anyone involved in design or fabrication can benefit from a corner radius calculator. This includes UI/UX designers perfecting button shapes, architects designing curved windows, engineers in CNC machining creating fillets, and even DIY enthusiasts crafting furniture. A common misconception is that any curve will do, but using a calculated, consistent corner radius creates a more professional and predictable outcome. Using a dedicated corner radius calculator removes the guesswork.

Corner Radius Formula and Mathematical Explanation

The core of this corner radius calculator is a fundamental geometric formula derived from the properties of a circle. When you know the chord length (c) and the sagitta (s), you can find the radius (r) using the Intersecting Chords Theorem. The theorem leads to the following powerful formula:

r = (s² + (c/2)²) / (2s)

Here’s a step-by-step derivation:

1. Imagine a full circle with the arc segment. Draw a diameter that passes perpendicularly through the chord.
2. This diameter is split into two parts at the chord: one is the sagitta (s), and the other is (2r – s).
3. The chord is bisected by this diameter into two halves, each with length (c/2).
4. According to the Intersecting Chords Theorem, the product of the parts of one chord equals the product of the parts of the other. Thus: `(c/2) * (c/2) = s * (2r – s)`.
5. Simplifying gives `(c/2)² = 2rs – s²`.
6. Solving for r, we get `2rs = (c/2)² + s²`, which leads to the final formula used by our corner radius calculator: `r = (s² + (c/2)²) / (2s)`.

Variables Table

Variable	Meaning	Unit	Typical Range
r	Corner Radius	px, mm, cm, in	0.1 – 10,000+
c	Chord Length	px, mm, cm, in	1 – 5,000+
s	Sagitta (Arc Height)	px, mm, cm, in	0.1 – 1,000+

Practical Examples (Real-World Use Cases)

Example 1: UI/UX Design

A UI designer is creating a mobile app and wants a button with a subtle, pleasing curve. The button is 200px wide (the chord length), and they want the curve’s depth (sagitta) to be 8px. By inputting these values into the corner radius calculator, they find:

• Inputs: Chord = 200px, Sagitta = 8px
• Output Radius: The calculator computes a radius of 629px. The designer can now use `border-radius: 629px;` in their CSS for a mathematically perfect, smooth curve, rather than just guessing a value.

Example 2: Woodworking Project

A woodworker is building an arched headboard for a bed. The headboard span is 150 cm wide (chord), and they want the arch to rise 20 cm at its peak (sagitta). To create a jig for their router, they need the exact radius of the circle that forms this arch.

• Inputs: Chord = 150 cm, Sagitta = 20 cm
• Output Radius: The corner radius calculator determines the radius to be 150.625 cm. The woodworker can now use a trammel set to this radius to draw a perfect, large-scale arc for their cutting guide.

How to Use This Corner Radius Calculator

This tool is designed for ease of use and accuracy. Follow these simple steps to get your results:

1. Enter Chord Length (c): In the first input field, type the length of the straight line that connects the two endpoints of your desired corner curve.
2. Enter Sagitta (s): In the second field, enter the height of the arc. This is the distance from the center of the chord to the highest point of the curve.
3. Read the Real-Time Results: The calculator automatically updates. The primary result is the Corner Radius (r), displayed prominently. You will also see intermediate values like Arc Length and Sector Angle.
4. Analyze the Data: Use the results table and dynamic chart to understand the relationship between your inputs and the resulting geometry. This makes our tool more than just a calculator; it’s a design aid.

Key Factors That Affect Corner Radius Results

The output of a corner radius calculator is sensitive to several key factors. Understanding them helps in making better design and engineering decisions.

• Chord Length: A longer chord, for the same sagitta, will result in a much larger radius, creating a flatter, more gradual curve.
• Sagitta (Arc Height): This is the most influential factor. A very small sagitta relative to the chord length leads to an extremely large radius (a very slight curve). Conversely, as the sagitta approaches half the chord length, the arc approaches a semicircle, and the radius gets closer to the sagitta value.
• Ratio of Sagitta to Chord: The relationship between these two inputs dictates the “roundness” of the corner. A small ratio results in a subtle curve, while a larger ratio creates a more pronounced, circular corner.
• Unit Consistency: Ensure both inputs use the same unit (e.g., pixels, inches, cm). The calculated radius will be in that same unit. Our corner radius calculator assumes consistent units.
• Measurement Precision: Small errors in measuring the chord or sagitta, especially the sagitta, can lead to significant differences in the calculated radius. Precise measurements are crucial for accurate results.
• Application Context: The ideal radius depends on the context. In UI design, you might want a visually pleasing ratio (learn about UI design principles). In mechanical engineering, the radius might be dictated by the size of a tool bit, requiring you to adjust inputs to match a required radius.

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; it is always twice the length of the radius. This corner radius calculator computes the radius.

2. What units should I use in the calculator?

You can use any unit of length (px, mm, in, cm, etc.), as long as you are consistent. If you enter the chord length in pixels, enter the sagitta in pixels too. The resulting radius will be in pixels.

3. Why is my calculated radius so large?

A very large radius results from a small sagitta relative to the chord length. This indicates a very flat, subtle curve. This is a correct and expected geometric result, often seen in architecture and road design. If you need a smaller radius, you must increase the sagitta.

4. How does this relate to the CSS `border-radius` property?

This calculator provides the ideal mathematical value for a circular radius. You can use the output directly in your CSS `border-radius` property to create a perfect circular corner rather than guessing a value. For a button 200px wide, you might think `border-radius: 100px;` makes a semicircle, but this tool helps you define much more subtle and specific curves. Check our aspect ratio calculator for more design tools.

5. Can I use this for an arc greater than a semicircle?

No, the formula used by this corner radius calculator is intended for arcs that are a semicircle or less (i.e., where the sagitta is less than or equal to half the chord length).

6. How do I measure the sagitta of a physical object?

First, lay a straightedge across the two endpoints of the curve to represent the chord. Then, use a ruler to measure the perpendicular distance from the center of the straightedge to the highest point of the curve. That distance is your sagitta.

7. Is a “squircle” the same as a circular radius corner?

No. A squircle has a continuously changing curvature, making it appear smoother than a standard circular radius corner, which has a single, constant radius. This corner radius calculator is for true circular arcs. You can find more on this in our guide to advanced CSS shapes.

8. Why use a corner radius calculator instead of just a visual tool?

Visual tools are great for approximation, but a corner radius calculator provides mathematical precision. This is critical for engineering, manufacturing, and creating consistent, scalable digital designs where values must be exact and replicable.