Carpentry Math – Drawing a Circular Arch

I must admit that I find a certain satisfaction in the geometry that pops up in general carpentry. Fine Homebuilding has a nice video on drawing a circular arch that uses a basic geometric construction. This is something that I expect to be doing soon so I thought I would document it here. The interesting part to me is the basic geometry. Basically, the construction uses a tape measure as a compass. I will just add the following figure, which shows the basic approach.

Basic Arch Construction

Basic Arch Construction

This discussion focused on using geometric methods to determine the radius. You can use algebra to determine the radius. The following figure illustrates how to go about the derivation and provides an algebraic result.

Algebraic Derivation of the Radius

Algebraic Derivation of the Radius

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2 Responses to Carpentry Math – Drawing a Circular Arch

  1. Tim Carlson says:

    Hi, my names Tim. I had a geometric equation years back from by boss. He passed. I lost it. Can’t find it anywhere! It was something like c squared divided by (8h + h/2). Sorry! Some thin like that. I’m a trim carpenter that was very good in my time but since then doing remodel. I’d love it if someone could help with this. If you have any advice I’d love it. I’m not a math guru. I am clean at what I know and want to get back to it. Thank you, Tim Carlson

    • mathscinotes says:

      Hi Tim,
      Thanks for asking a question. I am guessing the formula your boss used is the same as I derived above, just simplified into a different form. Here is what I get when reformated my formula above.

      $latex \displaystyle r=\frac{{{{{\left( {\frac{c}{2}} \right)}}^{2}}+{{h}^{2}}}}{{2\cdot h}}=\frac{{\frac{{{{c}^{2}}}}{4}+{{h}^{2}}}}{{2\cdot h}}=\frac{{{{c}^{2}}}}{{8\cdot h}}+\frac{h}{2}$

      I used the variable name w for the width of the opening. Your boss used c. Otherwise, it is the same thing.

      If I am not clear, please ask another question. I love carpentry math. It is what started me on my engineering journey.


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