Radius of an Arch

The Radius of an Arch calculator computes the radius (r) of a circle that would trace an arc of a certain chord length (c) and at a certain height (h).

INSTRUCTIONS:  Choose units and enter the following:

• (c) Length of Chord (see diagram)
• (h) Height of Arc from Chord to Highest Point

Radius of Arch (r): The calculator returns the radius is in inches.  However, this can be automatically converted to numerous other length units (e.g. centimeters or yards) via the pull-down menu.

The Math / Science

The Radius of an Arch calculator computes the radius of a circle that can be used to mark an arch based on the cord length of the arch and the height of the arch.  Arches on buildings and arched railings can be built using this formula.  The following is a video (YouTube) that shows the use of this equation to build an arched railing on a porch. CLICK HERE.  This can also be used to trace lines for painted arches as in the photograph of arches painted on a barn door (Special thanks to my friend Austin).

The formula for the Radius of an Arch is:

 `r = c^2 / (8*h) + h/2`

where:

• r = radius of arch
• c = arch chord length
• h = height of arch

HOW TO GUIDE

Let's use my neighbor's sliding barn doors as our working example. But you'll see that it works for any application. 

1. First, measure how wide you want your arch.  This is the Length of the Chord (c) in the diagram, but it's the width of the white sections in the barn door way. 
2. Then, choose the drop you want from the apex.  This is (h) in the diagram.  On the doorway, it looks like the drop is about a foot.
3. With these measurements, you can use this calculator to compute the radius (r) of the circle that will be your guide.
4. Next, cut a piece of string or mark the string on a plumb-bob to the length of the radius (r).
5. Then, calculate and measure where the mid-point from the ends of your arch.
6. Then go to the apex of where you want the arch, at the highest point in the middle and lower your string straight down.  This is where the plumb-bob works well. The bottom of the string will mark your center point of the circle; mark it.
7. Then attach the end of your string to that center point with a tack or screw.
8. Then extend the string and let it work as a compass for you to draw your arc from side to side.

Viola! You're done.  You now have a perfectly rounded arch drawn on your material, and you can paint or cut with precision.

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