How to Calculate the Radius of an Arch from Width and Height

When framing curved entryways, bending crown mouldings, or building brick masonry arches, one fundamental geometric challenge arises on almost every jobsite:

You know the opening width (Span) and the desired height (Rise), but you need to know the exact Radius (R) to swing a compass or cut your plywood templates.

In this guide, we will break down the exact mathematical formula, explain how the intersecting chords theorem works, and walk through real-world carpentry examples with fractions.

---

The Master Chord Theorem Formula

For any circular or segmental arch, the radius R is derived directly from the Intersecting Chords Theorem (also known as the Sagitta Theorem):

R = (W² + 4H²) / (8H)

Where:

• W = Total Horizontal Span (Clear Opening Width)
• H = Vertical Rise (Arch Height from Springline to Crown)
• R = Radius of Curvature

---

Step-by-Step Calculation Example

Let’s calculate the radius for a standard interior room divider:

• Span (W): 48 inches
• Rise (H): 12 inches

Step 1: Square the Width and Height

• W² = 48 × 48 = 2304
• H² = 12 × 12 = 144
• 4H² = 4 × 144 = 576

Step 2: Sum the Numerator

• Numerator = 2304 + 576 = 2880

Step 3: Compute the Denominator

• Denominator = 8 × H = 8 × 12 = 96

Step 4: Divide

• R = 2880 / 96 = 30.00 inches

Thus, a 48” wide × 12” high arch has an exact radius of 30 inches.

---

What If the Radius Exceeds Your Workshop Space?

For very shallow arches (such as a 12-foot wide patio opening with only a 6-inch rise), the mathematical radius can exceed 36 feet! In such cases, swinging a physical compass is impossible indoors.

Instead, use our Offset-Coordinate Mark-Out Table to calculate vertical heights every 2 or 3 inches directly along your plywood sheet.