The Degree to Which Your Angle Lies

Here’s an old school carpenter’s (or landscaper’s) method of laying out a line, such as a foundation form or a hedge row, to a specified angle. The tools needed are simple, primitive even: A length of rope marked at a certain distance and a 10′ pole marked in 1′ increments (i.e. the once ubiquitous carpenter’s 1o’ pole). Or you can join the 20th century and use a tape measure.

degrees-to-pole

Let’s jump right in and lay out an 8° angle from a baseline. The drawing above is pretty self explanatory, but I’ll explain it anyway in my hopefully not too pedantic step-by-step fashion:

Step 1:  Establish the baseline (via a stretched string) and set a pin (a sharpened stick works) at the focal point where the angle will converge.

Step 2:  Make a loop at the end of a non-stretchable rope (i.e. avoid nylon) and run it out along the baseline from the base pin. Measure out 57′ 2-1/2″ from the pin along the rope and make a mark with a Sharpie or tie on a piece of string. Also, set a pin at the baseline at that distance.

Step 3:  Now arc the rope away from the baseline in the direction you want to lay out the angle.

Step 4: Set the base of the 10′ pole at the baseline pin and orient it to the rope. When the 8′ mark on the pole passes over the mark on the rope then the angle to the baseline is (drum roll) 8°.

So how does this work you might ask? As my friend Joe Youcha of buildingtoteach.com explained to me: “The answer is buried in the math we were all injected with in grammar school.” We were all told about the “transcendental number” called “pi” which when inputted into your calculator would provide you with either the circumference of a circle based on its diameter or vice versa.

Artisans of antiquity, however, had no knowledge of the decimal number pi. In fact, decimal numbers in general had not been described in detail in the Western world until the late 1500s by the mathematician Simon Stevin. But artisans did have an excellent working relationship with the straightforward (non-cendental?) proportional ratio system. In the case of the relationship of the diameter of a circle with its circumference, they would just step out the diameter into seven segments and know that 22 of those segments would, to a high level of accuracy, give them the length of the circumference. Good enough for government work (such as the Parthenon) as they say.

Because we apparently need to work with degrees (probably because the architect speced out the angle in degrees instead of the length of a chord as they would have in antiquity), we would need to know what number of segments the diameter would be if the circumference were stepped out to 360 segments. That number is, of course, an arbitrary but widely accepted convention since Babylonian times as a convenient way to divvy up a circle. We like it as it can be evenly divided by so many whole number divisions – though for a time Europeans were quite fond of 400 degrees.

But I digress; back to how it works: If you go to the trouble of physically stepping out along the circumference of a circle with dividers, you’ll discover that when 360 segments do the trick, 114 and 5/12ths of another segment will define the diameter. Of course, using al-Jabr (given to us by the Islamic mathematicians), we can quickly solve for this result using an algebraic equation to solve for an unknown.

For this purpose we’ll use half of the diameter segments – fifty seven and two and one half twelfths – to lay out the radius length on the rope. The bottom line: We find that a radius of 57 feet, 2-1/2″ produces a circumference length of 360 feet. So for every foot we swing the arc, we produce an angle of 1°.

— Jim Tolpin, ByHandandEye.com (Drawing by Andrea Love)

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24 Responses to The Degree to Which Your Angle Lies

  1. David Evers says:

    Only good for small angles where the chord and circumferential distance are similar.

    • colsdave says:

      It’s valid for larger angles than one might think depending on the tolerance for error. A 1% angular error doesn’t occur until the chord equals 28 feet where the angle is 0.29° larger than intended. As the desired angle increases from here, the error accelerates rapidly using this method but we have other methods to accurately define larger angular segments as Hound demonstrates elsewhere. By combining those methods with this chord method, one could define most any desired angle with a respectable degree of precision.

      Incidentally, 180 feet divided by π is 57 feet 3-1/2″.

      • Mark Dennehy says:

        Not so much. Remember your cosine law : c^2=a^2+b^2−2abcosθ.
        Let’s say you tried this trick for 20 degrees. Cos 20 is 0.937. Let’s use 100 feet as the baseline just to keep the numbers nice for now; so 100^2 + 100^2 – 2.100.100.Cos(20) = 1206.15 and that’s your distance squared, so the distance you need to mark off on the far end for a 20 degree angle is 34.7 feet (if you did this with 57 feet 3-1/2″ you’d still get the same error percentage).
        You measure off 20 feet and you don’t get 20 degrees, you get 11.47 degrees, which is an error of 42%. And the drop-off is quite steep. Up to about 10 degrees you’re okay; every degree after that the error climbs further than the degree before.

        Honestly, everyone’s got a smartphone these days or a calculator; if you want to use this trick, take 15 seconds and work out the right length to use at the far end with the cosine law.

        • colsdave says:

          I used a spreadsheet for my calcs so I could easily use formulas to compute multiple angles to determine the rate of decay. I did not use the cosine law for my calcs; instead I used standard route surveying formulas since that is what I am used to using and that basically is the task in the example. Both are valid approaches for number crunching.

          I got my formulas from here http://www.esf.edu/for/germain/Horizontal%20Curve%20Formulae.pdf which has a good accompanying diagram. Since we wish to test the hypothesis that the LC is approximately equal to the subtended angle for a circle of 360′ circumference, I used the formula for the Long Chord where LC=2*R*sin(Δ/2).

          Rearranging the formula to solve for Δ gives us Δ=2*asin(LC/(2*R)). Since the Radius (180/π = 57.296′ for a circle of 360′ circumference) and Chord (lets use 20′) are our givens, we have:
          Δ = 2*asin(20/(2*57.296)) = 2*asin(0.17453) = 20.10 degrees.
          This gives us an angular error of +0.51%. Not too bad in my eye.

          Now, since the chord for a given arc varies in a linear fashion with the radius, we can easily calculate the chord for the same angle on any other arc using the ratio between the two radii where LC2 = LC1*(R2/R1). Using the above LC1=20′ and R1=57.296′, we can solve for LC2 where R2=100′:
          LC2 = 20*(100/57.296) = 34.906′
          Therefore, if I wish to establish an angle of approximately 20 degrees, I can measure a chord of 20′ on an arc with a radius of 57.3′ (circumference of 360′) or I can measure an chord of 34.9′ on an arc with a radius of 100′ (circumference of 628.3′). If I measure a chord other than 34.9′ on a circle of radius 100′, I will get a different resulting angle.

          I can calculate a different chord for any circumference but the numbers for *this method* work best where 360 linear units are used for the reference circle circumference.

          Usually, the smartphone or computer is not the challenge, it is the user. After looking at this for quite a while this morning, I think the blunder here is yours, not mine (this time…).

          • joefromoklahoma says:

            Yikes – look out, weeds ahead. It seems like what we are asking is “to what extent can we use the length of a chord to approximate the length of the curve for a given included angle?” The length along the curve is a function of the radius and the included angle in radians – for our example (because the circumference is 360 units), that’s a unit of length for every degree. So, can we lay out a story pole with this same length and use it as a chord to subtend the same angle? Nope – chord length is always shorter than arc length (don’t believe me – ask Archimedes), so the subtended angle will always be larger (a story pole of arc units will be longer than the necessary chord). Is it close enough for a good approximation? Yes, for small included angles – the smaller the angle, the better. For the 20° example, we’d be off by around 0.10 – feet or degrees. For 45°, we’d be off by about 1.15 units, probably still good if we’re just heading off towards Aunt Nancy’s (not quite so good if it’s the legal description of highway right-of-way). Like you say – depends on your tolerance for error. Don’t use this to layout a right angle.

            The surveyors know all about this because there are two ways to define the included angle of circular curves. In one, the “degree of curve” is defined by a radius and a 100’ arc length; in the other, the “degree of curve” is defined by a radius and a 100’ chord length. These are not the same curve – for a given radius, the included angle must vary; for a given angle, the radius must vary. The chord definition is used for most railroad work and by the military; the arc definition is used by most state highway departments. No judgment of either method – but consistency is essential.

            The thing is that for most large-scale layout, it’s impractical to stake the curve by locating the center and swinging an arc. Such curves are laid out by staking the ends of a series of chords. In highway work, these chords are generally selected to correspond to 100’ stations along the alignment, so the angles and chord lengths are specifically calculated to place the endpoint of the chord on the circumference of the curve (this was my understanding of Jay’s comment – finding an exact chord length – not an arbitrary chord length – that, when combined with a given radius and given angle, would define a point on the circumference). These chords are never longer than 100’ – the length of the chain – and can be substantially less in order to reduce the discrepancy between arc length and chord length to fall within the allowable error for taping.

            This is all iron transit and chain surveying – a lot’s changed in 45 years – but it is most definitely “geometry” – earth measuring. It’s also a textbook example of “the more you run over a dead cat, the flatter it gets” – my apologies.

            • Great reply Joe…I completely left out and forgot all about following my Survey collegues around during and after the Marines…This is “old hat” to you folks…To see someone else mention “chains” here was refreshing as not to many of us still see, and use them much these days.

        • jenohdit says:

          They problem with your analysis is that you have substituted an arbitrary baseline of 100 feet when the original length of the baseline, 57 feet 3-1/2″, is essential to the method proposed.

          This method draws an isosceles triangle. If 2 triangles isosceles triangles with bases of the same length have legs of different lengths then the angle between those legs will be smaller in the triangle with the longer legs. That’s why you only get 11.47 degrees.

          The error in Tolpin’s method will be in describing an angle slightly larger than intended because the chord is shorter than the circumference of the circle it intersects.

  2. Art Shaw says:

    The same thing can be done at any scale. A few years ago I needed to lay out a circle in which I had to situate things accurately. I wanted to make a degree easy to measure on the circumference, so I took 1/4″ to be 1 degree, which meant that the circumference was 360 X 1/4 = 90″. Then I divided 90 by pi and came up with a diameter of 28.648″ and a radius of 14.324″. When I laid out that circle on my plywood,each degree was 1/4″ at the circumference, which I could step off with dividers.

  3. Hmmm…??

    Hi Jim,

    To validate just a little bit, I’m not an expert (per se) on this but I do teach Trig as it applies to traditional layout methods, and also empirical layout methods of the…”old ways.” Being one that built my first timber frame without touching a tape measure at age 19, and having learned the craft…in the old ways…(from Amish Barnwrights…i.e. story poles, dividers, trammel, chains, rope and pacing/dead reckoning, etc.)…I could only follow so far before things seemed a tad off from what I was taught…???

    From your example given…and to achieve an actual 8° slope…the Story pole (aka ubiquitous Carpenters 10′ pole) would be held at a 90° to the string/rope, and the distance marked out would be 56′ 11 – 1/16″ to achieve 8°…not 57′ 2 – 1/2″.

    What you have (back checking with trig, and my memory of similar methods taught to me) would provide ~7.96°. I totally agree (and understand) this might be splitting hairs (or pedantic…which I can be…ha, ha) yet I am trying to make sure I haven’t missed your point and/or understanding the method you are trying to share…

    With 56′ 11 -1/16″ and the string/rope NOT moved at all…8′ mark at 90°…provides ~8.00015°

    Where it seems our two methods differ (form my original teaching as I remember it…??) is the rope/string/chain does not move and the 8′ mark on the Story Pole will designate the 8° that is desired…

    Perhaps another angle example (or several?) besides 8° would dial in the difference between what I remember of this method, and your given example?

    There are a number of modalities within this theme…

    Sorry if I missed something or have confused things…GREAT post!!!

    • colsdave says:

      I think your misunderstanding arises from your assumptions based on your timber framing experience and calculating for a slope (which is a vertical angle). When we measure vertical angles, we can make convenient and accurate reference to the concepts of plumb and level for measurement using the primitive tools of a plumb bob and a spirit level of some sort. Level and plumb are primary characteristics of a timber frame for a structure; other angles are less so. In any case, the presumed builder can more easily determine desired slopes using rise and run instead of a degree slope, so that is what the designer typically supplies. No device is needed to measure angles – the builder just needs to use a plumb bob beside his 10-foot-pole as a reference to keep the pole plumb to achieve the desired accuracy after using the level to transfer the starting point elevation from the toe of slope to the bottom of the pole. If the slope is off by 1%, that is only 1/8″ in 12″ and our humble timber-framer can probably do better with a little care. (Either way, the cows in the TF-barn don’t mind and the folks in the non-electric TF-house likely don’t either since things move over time anyway.)

      In the example in the original post, we are establishing a *horizontal* angle between two lines laying on the ground (or drafting table). The concepts of plumb and level and their related tools are no longer useful to us to measure this angle. Instead, we can use circles and lines as our primary references along with our 10-foot-pole.

      Let say we are anticipating the arrival of sacred Hawaiian emus on their breeding migration and we wish honor them by building a timber-framed temple for them to rest in and care for their garb (emu muumuus require a suprising amount of attention). It has been determined that this temple (with an attendant pool where the emu can bathe and pass gas) should be positioned with one corner at a certain point along the edge of a small straight path (along which we scatter the remains of our fresh-brewed coffee ritual) and with the face of the building rotated 8 degrees counter clockwise from the path edge. (The feng shui of sacred emu temples is beyond the scope of this post but, considering the number of chili suppers they are guests at during their perilous perigrinations, the feng shui concerns are considerable.)

      Our Humble Timber-Framer lacks any means of directly measuring an angle but he is wise to mysteries of the arcane 360′-circumference circle. From the designated point (the base pin in the original post drawing), OHTF stretches his revered rope along the edge of the path the mystical distance of 57′ 3-1/2″. Then, using the revered rope and leaving the straight and narrow, OHTF swings an arc counterclockwise scribing a curved line at that distance on the hallowed grounds. Now, according to the ordained liturgy, OHTF solemnly places the zero end of the consecrated 10-foot-pole at the point where the arc meets the path and carefully aligns its 8-foot mark at that point where the edge of the pole crosses the arc. The face of the temple can now be established by using this point and the base point on the hallowed grounds to properly honor the windy wanderers.

      It is important to note that the consecrated 10-foot-pole does not lie perpendicular to the path – instead, it lies spanning the requisite chord on the arc for the angle deflected (8 degrees here for our emu temple) and lies at an angle to a line perpendular to the path equal to one-half of the primary deflected angle. This proves that following the straight and narrow path through the hallowed grounds does not always put you where you should be to fitly facilitate the fecund flightless flatulent fowl. Hence the subtleties of the sacred 360′-circumference circle serves to confound the casual kibitzer.

      • Hello ColsDave,

        Thanks for such a detailed reply…Looks like this post made an impact on many…

        I may have missed some of the post’s intent…or…I approach it from a different “angle” (pun…ha, ha) as it looks like many did.

        I was not just thinking timber framing, but stone sculpting, stone work, and even a bit of rigging and shooting arts as well. The list is long that these types of…Maths…can help with particularly…traditional empirical applications. Many solutions to some common challenges.

        “…Level and plumb are primary characteristics of a timber frame for a structure; other angles are less so…”

        Sorry…been doing it for over 40 years and that statement isn’t accurate at all…Timber framing (as a traditional art) is about so much more than just “level and plumb” especially when you get into forms lie Minka, Hanok, Chise…not to mention all the complicated compound roof forms…

        Don’t use rise/run to often much anymore…??…mainly degrees yet the two are so fully interchangeable conceptually and in application.

        I would also suggest that Timberwrights- Barnwrights are very much concerned with precision much greater than 1/8″! We are working on a structure now (for example) that is over 45 meters long and our tolerance in layout is 0.5 mm over that distance. Do we achieve it? Hell no!, but in this craft one (if they are good at it that is) can not tolerate aiming any less accurately…”Aim small…miss small,” applies in the shooting arts and timber framing alike.

        In this post example…be it a hedge foundation, stone wall, or anything else…flat on the ground or perpendicular two it, the method described will work…the method I was taught does as well yet perhaps a bit more accurately and just as empirically. There are parts/details missing (for brevity in the conversation of course) and more traditional examples as well…

        • colsdave says:

          “We are working on a structure now (for example) that is over 45 meters long and our tolerance in layout is 0.5 mm over that distance.”

          I suspect you are using instruments more sophisticated than a plumb bob, hand-held spirit level, rope and a 10-foot-pole in pursuit of that precision. Are there any pictures or renderings of this structure that you can point us to?

          • jenohdit says:

            Incredible control of temperature and humidity too it seems. Yes, pictures please!

            • Hello Colsdave and Jenohodit,

              I will validate a bit more in detail to perhaps reflect the way I approach most work..be it stone, earth, timber, textile, or other…

              I started earnestly following the many arts within my family’s different trade crafts/arts at 13, then begin at this time to apprenticed (on and off) to Old Order Amish Barnwrights, til I was 23. I didn’t touch a tape measure or instrument of modernity within timber framing till after having been around many traditionally facilitated frames; then building my first at 19 in the same manner.

              I still do not employ…”sophisticated instruments”… to rely on for my primary layout and work…other than perhaps a modern (metric) tape measure to create the story poles/tapes for a given frame design, yet even then dividers to “pace off” a standard unit is not rare either. As such, a laser level may be brought in to speed up finding a…”rough zero point”….on naturally shaped stone plinth and post foundations members, however (!!) a Water Level is all I would ever trust to give me the marks I build from. So I may design now in CAD to stay in pace with business of the day…but we still rely on hand drawn paper drawings in the day to day work, and our tooling to finish is all by hand almost in the entirety.

              So to your query direct..DO NOT…rely on..”sophisticated instruments”…and most certainly do LOVE!!! (and need!!) our plumb bobs, (I have two next to me on a timber right now!), hand worked leveling devises, chain/rope/string and most definitely our very much beloved…Story Poles (aka 10 foot pole) …because without that most preciously cherished of Wooden Stick, we would be certainly completely lost and utterly dumbfounded!!!!

              “…Are there any pictures or renderings of this structure that you can point us to?”

              Anyone is more than welcome to contact me on these subjects for both academic discussion on the topics or related business. The frame in reference can be followed (it is my 2016 Tithing Project) at: http://www.facebook.com/Menomonie-Farmers-Market-132792951987/

              And again…much thanks for such a great post and discussion of traditional mark and measure modalities!!

          • Building something at CERN by chance, with those tolerances?

            • CERN’s tolarance (like NASA’s) are much greater than mine…ha, ha…:)

              Timberwrights in general are an exacting bunch…at least those with some time under their belt, pride in there craft and knowledge about traditional green (aka wet) wood working.

              Like I shared earlier, you have no choice but to “aim small” or your joints will gap worse than the checking in the wood itself…The joints should melt into each other if the layout and cutting is performed well…whether the joinery is of a set geometric shape, or live edged morphing type…

    • jenohdit says:

      Using the law of cosines gives an angle of 8.01877 degrees for the figures in the drawing above. The error will be in increasing the angle by a miniscule amount rather than decreasing. The chord is shorter than the arc length.

      • jenohdit says:

        colsdave’s correctly notes – “Incidentally, 180 feet divided by π is 57 feet 3-1/2″”

        Using his corrected figure for the baseline results in an angle of 8.00709 degrees, which is good enough for most government work (although maybe not in Switzerland).

  4. I learned this as a carpenter’s trick for use with a stick folding rule, but a little differently.

    Drill a small hole in your folding rule at 57 9/32 ” ( or 573 mm if you use a metric rule) and use a small Brad in it as the radius point to scribe the radius of a circle on, say, a piece of plywood with a pencil ( or use trammel points with that radius to scribe the arc) the BEND the folding rule along that radius and each number increment on the rule will be one degree to a very high accuracy even at large angles, unlike your example which only is accurate at small angles. The key is that the rule has to follow the arc.
    If you have less room, 28 5/8″ will work, with each degree corresponding to 1/2″, but a weak rule might break.
    ( each of these imperial numbers is a little small, allowing for the width of the pencil point. The metric number will be close enough.)

    You could, I suppose, use a tape measure, but that’s harder to keep the whole tape along the radius, which is key to the method.

    And yes, I’m dating myself. BITD every tradesman had a stick rule and I still use mine.

  5. jtolpin says:

    I think the most accurate method for any angle would be to draw the arc and then step off the 1/360 segments along the curve–each one degree. Impractical if you aren’t working on a flat surface or to a large scale however.

  6. dkmarion says:

    57′ 3 1/2 inches, 360/(2*PI) = 57.295…feet

  7. *(i.e. the once ubiquitous adventurer’s 10’ pole)

  8. joefromoklahoma says:

    This is a great trick and reflects a practical understanding of the ratios – rational and irrational – contained in the circle. I had not thought to mark my folding ruler to incorporate this, so I doff my hat and will do do later this AM.
    Seems like most of the discussion revolves around how to practically lay out the “chord” – perpendicular to the base or perpendicular to the revolving radius – one works the sine, the other works the tangent. This all works well at small angles because the sine and the tangent converge as the angle decreases. Back in slide rule days (rolls eyes), we had S scales for sines and T scales for tangents and ST scales for values below 5.74 degrees (which was pretty arbitrary and mostly based on sin5.74=0.10001 which was a good place to start the new scale). This rule still works above 5.74 degrees for some arbitrary range and then pretty well falls apart (the less you are plotting a course to the moon, the greater that range). Like Jim sez – the more you can use this to derive an small chord increment and step the angle off with that, the better. Cool in the shop, less so on ungraded terrain.
    I, for one, am looking forward to a second volume of By Hand and Eye that incorporates the geometry of the right triangle – trigonometry. You want ratios??? That’ll have some ratios. Great post, great comments, thanks!!

  9. Somewhat off topic, but when using a calculator to calculate an angle I lean more towards using TAN in stead, since it is always at a right angle to the baseline, hence easier to check and mark precisely. Beyond 50-60 degrees it becomes more cumbersome. Some would even argue 45 degrees (1:1).

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