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Circular-Based Arches – Part 1: One-Centered and Two-Centered Arches

I’ve toured a lot of historic homes and seen some extraordinary arches—door jambs, windows, passageways. In reading about historic architecture, especially Gothic and colonial styles, I’ve come across some beautiful arch work. But those once-common elements are not often incorporated into millwork today. Sure, sometimes the carpentry techniques are more difficult, and too costly, but the problem I’ve recognized is more one of design.

Common circular-based arches (Note: Click any image to enlarge)

Arches in modern homes often seem slightly off—there’s frequently something wrong with them, particularly when you compare arches built in homes today to historic designs. I couldn’t put my finger on the problem, so I started researching arch designs in pattern books and on the Internet. What I discovered is more a problem of communication than technique. Mixing arch designs—like this segmented entry door jamb and 3-centered stone arch—never works (see photo, right).

Combining a group of openings with segmental jambs can look awkward if the spring lines are at different elevations, if the tops of the arches vary in height, or if the spans are significantly different (see image, below, click to enlarge).

Segmental Openings

And segmented jambs can look even worse if keystones are used improperly. Remember, you can only put a keystone in one and only one spot—at the apex of the arch (see “Parts of an Arch,” below).

And another thing . . . segmented radius arches do not look good when they’re decorated with classical head details. Doesn’t there appear to be something missing in both of the pictures below? Yes, there is—structural support and a defined point of termination.

Certainly, there are a lot of builders and architects who aren’t reading Get Your House Right! But the real problem I found was with instructions for laying out arches—they are all terribly outdated! In fact, almost all of the information we use today has been collected and re-printed from books that were published over a century ago—illustrations filled with confusing text, multiple lines and intersections, usually with all the information compressed into one ink drawing (see image, right).

Publishing books a hundred years ago was prohibitively expensive: the cost of a single sheet of paper was so high that private letters were often written with the text running in both directions, just to save on paper. It’s no wonder book publishers never considered multiple, step-by-step illustrations.

But that’s not the case today—at least not for an e-magazine like THISisCarpentry! Now that we have paper-free publishing, it’s time to re-draw those old instructions.

The articles in this series are meant to provide a richer format for today’s “digital savvy” carpenter. There is still a fair bit of geometry involved, but fear not! All of these articles include Quick Reference Guides, or “cheat sheets” (downloadable PDFs), with step-by-step instructions for each arch layout.

Let’s get started:

Arch Basics

An arch is a structure that spans an opening and supports weight. Arches have been around for thousands of years, and were originally constructed out of stone. During the Roman Empire the engineering of the masonry arch was perfected and its structural element defined.

Even though decorative millwork doesn’t need to provide physical strength and support, it should do so visually. You can’t fool the eye. You might not know why, but something inside you will let you know if it doesn’t look quite right (just like the start of this next sentence!). It’s just like why choosing a terminating or supporting molding can make all the difference.

Parts of an arch (click to enlarge)

Important Terminology

Impost: The block set into a wall or uppermost part of a column or pillar, used to support an arch.

Keystone: A wedge-shaped piece at the apex of an arch that locks the structure together and allows it to bear weight. The shape of the keystone should always be related to the center point of the arc that makes up the arch.

Spring line: The line at which an arch begins—located at or above the impost.

Stilt: The elevation of the spring line above the impost.

Voussoir: A wedge-shaped piece used to make up the curved part of an arch.

Geometry Refresher

Because all of the following arch types are based on the circle, let’s review the fundamentals of circular geometry.

Anatomy of a circle

Important Terminology

Arc: A curved line that is part of the circumference of a circle.

Chord: A line segment joining two points of a curve.

Circumference: The distance around the perimeter of a circle.

Diameter: The distance across a circle through its center point.

Radius: The distance from the center point of a circle to its perimeter. Equal to one half of the diameter.

Point of Tangency (tangent point): The point at which the tangent touches an arc or circle.

Tangent: A line, arc, or circle that touches an arc or circle at only one point.

One-Centered Arches

Done correctly, segmental arches are versatile enough to even feel at home in a Craftsman style home.

Determining the radius of an arc for a given span and rise can be worked out with simple geometry, but if you have a construction calculator, you can find your radius with just a few key punches.

Here are the steps (I use BuildCalc on my iPad. If you use CMPro on your iPhone/iPad or Droid, the key locations are a little different, but steps are the same!):

1. Enter the desired span of the arch (48 inches in this example) and press RUN.

2. Enter the desired rise of the arch (6 inches in this example) and press RISE

3. Press the CONV key (when you press the convert key, the ARC key will change to the RADIUS key).

 

 

4. Press the RADIUS key to display the radius. (Note that at the completion of this calculation, BuildCalc's keys will revert back to their default settings. The Radius key becomes the Arc key again, as seen above.)

 

Finding the radius of a segmented arch

This function of a construction calculator can also be used if you need to find the radius of an existing inside curve.

1. Cut a straight piece of wood to a length that will fit inside the arch, and touch two points of its curve. The actual length of the stick is not important, but using a nice round number like 12 in. or 24 in. will make things easier. After cutting, measure and mark the midpoint along its length.

2. Place the piece of wood against the arch—it doesn’t matter where.

3. Measure the distance at a right angle from the top of the stick’s midpoint to the existing curve.

4. Enter that measurement into the calculator and press RISE.

5. Enter the length of the stick and press RUN.

6. Press the CONV key to change the ARC key to the RADIUS key.

7. Press the RADIUS Key.

For readers who don’t have a construction calculator, here is the formula you can use with a standard calculator. Unfortunately, you also have to convert any fractions to decimals.

 

Download the Quick Reference Guide for Segmental Arches

 

Two-Centered Arches

While Roman architecture is known for one-centered arches, two-centered arches are fundamental to Gothic architecture and form the simplest “pointed” arches.

The large main parlor window at Lyndhurst is framed by a two-centered arch.  
The lancet windows surrounding this tower are typical two-centered arches. The same motif repeats itself in a crenelated pattern across the porte cochere parapet walls. (Sells Mansion, Columbus Ohio)

Variations

There are many variations of two-centered arches, and each depends on the location of the center points. When the center points are located closer to the middle of the span, the arch flattens out; if the center points are located farther away from the middle of the span, the arch becomes sharper.

  The drop-arch on this fireplace, beneath a suspended hood, provides just the right amount of gothic flavor for an early 20th century arts-and-crafts home. (www.adamsonhouse.org)

The following Quick Reference Guide provides step-by-step procedures for finding the required arc centers and appropriate radii for a two-centered arch that must meet a specific height and width.

Download the Quick Reference Guide for 2-Centered Arches

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Note: A recurring step found in these geometric constructions is to draw a line perpendicular to another line’s midpoint. For simplicity, a square has been used in the illustrations, but the task can also be accomplished with just a compass/trammel and a straight edge, as shown in the following Quick Reference Guide.

Download the Quick Reference Guide for Bisecting a Line

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Don’t miss the next article in this series on Three-Centered arches, where the geometry gets a little more complicated.

Comments/Discussion

30 Responses to “Circular-Based Arches – Part 1: One-Centered and Two-Centered Arches”

  1. josh o

    Wow! Great article. Nice pictures, videos and text.

    90% of the arches I see out there bug the crap out of me. The worst offender is semicircular windows that NEVER have the arch stilted.

    Never knew how mixing arch types really doesn’t work.

    Thanks so much.

    Reply
  2. Alex M

    Great article! This is the stuff I love to read! So many times someone has a set of prints with all the drawings on them, or has already ordered a door or window and you are just stuck making templates and fabricating mouldings for arches that really don’t work together and you are left scratching your head wondering where alot of these designers and architects get their training.
    We often tell our clients that just because you can’t tell what it is that is wrong, doesn’t mean it’s right! You can’t fool the eyes!
    Fantastic job writing this article, great pics, vids and explanations. Can’t wait for part 2!

    Reply
  3. OKMrazor

    When I read the title, I could hear the music start to play. Once I started reading, I could hear the choir sing… and that’s a tune I’d join in anytime. Great article.

    Reply
  4. Craig

    I see many arches that are oval shaped on houses. Quite often arches on garage doors do not match the glass in the door.

    Reply
  5. Jesse Wright

    Todd,

    I feel like the Double rainbow guy every time I see your Sketchup animations. This article is amazing, and I hope that everyone reading especially the ones who are doing all the built-ins with arches, are paying attention! Thank you for this timely piece, looking forward to part 2.

    Double rainbow guy ” OMG OMG”

    Reply
  6. Jim Baldwin

    When calculating radius without the benefit of a construction calculator, I prefer to dummy things down and write it all down in a little book.. I don’t use “Rise and Run” since both words start with “R” and those terms (for stairbuilders) are already spoken for.

    My little lingo and memory crutch starts with a “C” for Cord and “A” for Arrow (as in bow and arrow) with “C” being the bowstring.(I thinks me thinks better in pitchers)

    Anyway…
    C / 2 (STO) / A X (RCL) + A / 2 = Radius
    or…
    Half the cord divided by the arrow, times half the cord plus the arrow, divided by two.

    I’ve used this formula so often that it really ought to be engraved on my tombstone..

    Anyway, very fine article and I’m looking forward to learning more about the “basket-handle arch”? (an elliptical arch that does indeed look like a basket handle) I’ve never heard that term before.

    Reply
  7. Lars Jensen

    A common arch error: flexing a stick to create the curved line. This does not provide a circular arch because the stick flexes more in the center than on the ends.

    Reply
  8. Dave

    Loved the first two photos, of wrongthink arches. Amazing that nobody has written this guide before. Well done!

    Reply
  9. Todd Murdock

    Thank you all for the kind comments!

    This project was a labor of love, and never would have been possible without the help and encouragement of Gary Katz. I would also like to thank Robert Walker, TiC’s managing editor, for his hard work and patience with me.

    I hope you enjoy the next two installments and find the information useful.

    Todd

    Reply
  10. Larry

    I’ve never run across the term “suspected hood” before. Please enlighten me.
    Lots of houses have strange proportions that go way beyond the arches.

    Reply
    • Gary Katz

      Neither have I! I think we created a new term. A suspected hood: one who enters a home and doesn’t like the arch-i-tecture. Actually, I think we meant “suspended.” :)
      Gary

      Reply
  11. Brent

    A very good resource and compliment to this article is: Traditional Construction Patterns (Stephen Mouzon). I just got back from Trento and Verona, Italy and I couldn’t find any design feature out of proportion; I was in fact blown away. My wife has all these pictures of me, not window shopping or sipping fine wine, but looking up at all the buildings. Very good article and compliment to our trip.

    Reply
  12. Mike Pelletier

    Good article on such a beautiful feature. A downside to our manufacturing prowness is that it has become very easy to do it wrong. I suspect historically, when it was a more deliberate and costly detail it was probably more thought out in the design stage. Now folks often just pick it out of a catalog and make it fit.

    I have never heard the “basket handle” term before, I like it. I look forward to the three centered arch discussion. I like the 3 centered arch over the true ellipse, particularly because it allows one to run all the profiles on a shaper with simple radius jigs- yet is comparably elegant.

    Reply
  13. Sam Marsico

    Thank you for taking the time to do this article. I am looking forward to learning the relationship/difference between the three-centered, “basket handle” arch, and an ellipse. I have encountered situations when working with wide passages where a true ellipse looks wrong due to the tight radii ends. I guess that probably means the opening was too wide for an arch in the first place :)

    Reply
  14. teeg merchant

    Thanks for the wonderfully informative article. As it happens, I have been tasked with fabricating some molding to fit above a bay window-not an arch but an arc none the less. I will certainly use the chord/rise/run formula that you show, so thanks again. Also, I have a nice biography of Fillipo Brunelleschi, the designer and builder of Il Duomo in Milan. The book has a diagram and description of a Gothic arch that is called a quinto arcuto as the layout is derived from 1/5 of the span. The methodology is the same as you describe. Whenever I am tempted to whine about not having some particular tool or skill or material i just look at a Gothic cathedral or a piece of 16th century furniture and go back to work.
    TGM

    Reply
  15. Doug Simmons

    Wow, just finished with two of the one centered arches, knowing that there was a formula out there. No way to get on the internet at the jobsite, so I used the trial and error method of trying different radius along the centerline until I found the correct one. One guy suggested the old “bend the stick” method, to which I said “no way”. That nite I found the formula and used it to do the second arch. Thanks a million for these articles.

    Reply
  16. Brady

    Todd –
    Great article! Thanks for including the downloadable cheatsheets — should be really helpful when trying to walk a crew member through how to create various arches on the jobsite.

    Like everyone else, I’m looking forward to your discussion on the 3 center/basket handle/elliptical arch (its a personal favorite!)

    -Brady

    Reply
  17. Wendy Ice

    Hi,
    Thank you for this valuable resource. I wonder if you might know the answer to a question…I’m a decorator working on a remodel on a 1910 house with some Craftsman influence. The architect has been wanting to insert one-center arched niches and they look wrong to me but I hate to bring it up without doing my research. I tend to associate this shape with homes of a slightly later vintage (1920’s) whereas I often think of gothic or tudor arches in the 1910’s. As much as I like them, I’d prefer to avoid arches altogether on this particular project—we have plenty of other great lines to echo from the original design. But if we use any, I’d like to make sure they are period appropriate. Thoughts?

    Reply
  18. David Baca

    Good Evening,

    I am designing a new Arts & Crafts bungalow for my family. I have (3) 3′-0″ wide arches leading to various rooms/hallway and (1) 11′-1″ wide arch between the kitchen and dining room. All (4) arches can be seen from multiple locations, however the small arches are not on the same wall as the large arch. Would you recommend that I keep the rise the same for all arches in this situation? I think it looks okay with a consistent or varied rise, honestly, but I’m a 30 year old structural engineer, not a 50+ year old carpentry expert.

    Thank you for your time.
    David

    Reply
    • Wm. Todd Murdock

      David,

      I’m actually half way between 30 and 50, and I certainly don’t have all of the answers… If any at all! :)

      The biggest problem I’ve seen in the arch designs used in millwork today is that they are used simply as decorative after thoughts. What could be a focal point of strength and visual support to the design ends up looking like just a curved detail.

      Keeping the heights of the arches in a room equal may be necessary because of existing conditions, but don’t sacrifice visual strength just for symmetry. Low rising segmental arches, splayed across a large distance, just tend to look too delicate. Using a depressed arch; like an elliptical, three centered, or pseudo three centered arch will keep the same rise, and still provide symmetry and visual strength for wider openings. Here is an example from Thomas Jefferson’s Map Room at Monticello. One room, two arches, same rise…

      Todd

      Reply
      • David Baca

        Well if Thomas Jefferson can mix arches…

        Interesting picture, Todd. I agree with your comments regarding long single circle arches looking delicate, but I think I’m okay with that look. I think what I’m trying to avoid is creating something that’s “wrong” without being aware of it until the drywall is up, at which point it’s too late. I tried attaching a JPG, but I received an error…email??

        Reply
  19. Glenn Luse

    Do you have instruction for a Pseudo 4-centered or a multicircular arch form?

    Reply
  20. Scott B

    I’m opening a wall between kitchen and living room. The finished opening will be ~14 feet wide. Ceilings are 8 feet. I’d like to add a gradual arch to the opening. Similar to the right-most example in the “Segmental Arches” graphic above. However, I’m not sure where the spring line of the arch should start. Should it be at standard door height of 80″ (6’8″)? Is there a building code that dictates a minimum height at the ends of the arch?
    There is a stairway at one end of the opening. So, I want to be sure there is no problem with headroom at the ends of the arch. But, because of the 8′ ceiling, I’m limited by how high I can go at the arch center and still allow for trim (prefer trim look over just drywall opening).

    Reply

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