Co-written by Mike Sloggatt
Around 2,500 years ago, a Greek philosopher we all met in high school named Pythagoras discovered a theorem that can make life easy for carpenters and contractors—if we just knew how to use it, and how to find right angles!
Most of us remember our ABC’s from high school, and we remember the Pythagorean Theorem, too, which applies to any 90-degree triangle.
| But we never learned how to use and apply Pythagoras’ extraordinary rule from a chalk board! Progressive carpenters know it’s never too late to learn; in fact, learning something new is the glue that bonds us to carpentry, and the jobsite is the perfect classroom. |
 (Note: Click any image to enlarge) |
Construction calculators make it easy for carpenters to use the Pythagorean Theorem on the jobsite, and in inches and feet! The calculator translates a, b, & c into Rise, Run, and Diagonal.
It also includes a “PITCH” key that allows you to enter or calculate the angles of the triangle using trigonometric functions. The key thing to remember about Pitch on a construction calculator is that it is always the angle opposite the Rise.
Maybe we call this a “right triangle” not just because it has a right angle, but because it’s the right triangle for solving almost all geometry problems…especially on the jobsite. Using the right triangle is easy: If we know at least two dimensions or one dimension and an angle of a right triangle, we can solve for the remaining dimensions or angles. Sometimes the biggest problem is finding right triangles and knowing how to use them.
Finding Right Angles in Foundations
Laying out foundations used to be a slow, tedious process. I remember my father’s foreman, Loren, used to carry a well-worn folded paper in his wallet with a list of 3-4-5 variables that my uncle had written out for him. That list started with 3′ x 4′ x 5′, and it went all the way up to 30′ x 40′ x 50′, in 2 ft. increments! Loren was proud of that paper and showed it to me when I was ten or twelve, when I watched him layout a foundation for the first time. Many carpenters still use the same method today.
A 3′ x 4′ x 5′ triangle is often too small to ensure accuracy for any size foundation, so carpenters usually choose the largest triangle possible for a given rectangular addition. Then they double check that the layout is square by measuring diagonals and laboriously moving corner points until the diagonals are equal. But all that effort is unnecessary. With a construction calculator, you cut directly to the right angle.
Laying out foundations is one example why old techniques aren’t always the best techniques. Today, carpenters frequently discover the hard way that many old methods are slower and less precise. With a construction calculator, laying out foundations is fast and exact. Simply enter the RISE and RUN, then press the DIAGONAL key. A carpenter working alone and holding two tape measures—one pulled along the 20′ Rise and another pulled along the 37′ – 8 13/16″ Diagonal—can simultaneously find the precise corner points and square up a foundation.
Finding Right Angles in Framing
Framing is another chore that a construction calculator can simplify and improve. Whether you’re framing a bay pop-out in a floor or a gable end, knowing your exact layout—along both the horizontal and the raked plates—and knowing the exact length of your studs or joists, reduces framing time by more than half, and ensures accuracy.
Most framers would project their joists across the corner of a bay pop-out, or measure each one individually, and they’d measure the layout perpendicular from each previous joist. But it’s much faster to see and use the right angle.
| The right angle is formed by the rim joist and the first joist. Even though you haven’t installed it yet, you know it’s going to be there. On a 30-degree bay, enter 30 on your calculator, then press the PITCH key. If the bay is 45 degrees, enter 45 and press the PITCH key. |  |
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If the joists or studs are on 16-in. centers, you’ll know two things about the right angle: the Pitch and the Run. Enter 16 in. and press the RUN key. |
| Press the RISE key to find the length of the first joist or stud. Remember, the RISE is always opposite the Pitch (and vice versa!). |  |
Here’s where the calculator really shines. Leave 9 1/4 in. on the display. To find the length of the next joist or stud, press the “+” key once, then press the “= ” key. The calculator will add 9 1/4 in. to itself when you press the “+” key. To find the length of all remaining joists or studs, don’t press the “+” key again! If you do that, you’ll be adding the new number in the display to itself and losing the decimal fraction in the calculator’s memory. Instead, press only the “=” key for each succeeding joist or stud!
Remember, the calculator is rounding off the actual decimal measurement to 9 1/4 in. If the measurement isn’t exactly 1/4 in. or even 1/16 in., the calculator will always round off to the nearest fractional measurement, eliminating any cumulative error (The fractional resolution preference on the calculator can be set from 1/2-in. to 1/64-in.). Note: Most construction calculators also include a “Rake Wall” function that can be used for these calculations, but it is beyond the scope of this article.
| Use the same sequence to layout the “diagonal” rim joist or top plate. Enter 30 and press PITCH, then enter 16 in. and press RUN, and then press DIAGONAL to find the distance along the rim to the first joist. |  |
To find the exact layout of the succeeding joists or studs, use the same procedure used for the joist/stud lengths—press the “+” key followed by the “=” key for the second layout mark, and only the “=” key for each succeeding layout mark!
Finding Right Angles in Finish Work: Cabinet Crown
Foundations and framing aren’t the only places where right angles occur.
I had no problem cutting all the crown pieces for these rectangular cabinets—I just added 1 in. for each overhanging side. But cutting the crown molding for the corner cabinet was another story. I cut all the pieces long, figuring I’d mark them for exact length in position on the cabinet. Of course, Mike pre-assembled the pieces, thinking they were all cut to the right length!
“What’s up with these?” Mike stood on the ladder, nail gun in hand, wondering why the assembly didn’t fit. “I couldn’t figure out the length,” I said. “I meant to mark those in place!” Mike responded, “But didn’t you see the right angle?!”
The cornice is made up of three pieces—the bead forms the base for the fascia and crown. The bead molding projects exactly 1 in. beyond the cabinet edge. Calculating long point measurements on the rectangular cabinets was easy—I added 1 in. to the cabinet’s side measurement for the side pieces, and I added 2 in. (one inch for each outside corner) to the cabinet’s front measurement.
But figuring out the long-point measurement on the corner cabinet wasn’t so easy. Rather than transferring lines back onto the inside of the cabinet and cutting from short-point measurements, it’s much easier and more precise to find the right angle.
The right angle in this example is imaginary—it’s not formed by framing or a foundation, but instead by the miter angle required for the corner cabinet (22 1/2 degrees), and the overhang of the bead molding.
| Enter 22 1/2 for the PITCH (remember, the PITCH is always opposite the RISE). Enter 1 in. for the RUN, and then press the RISE key. |  |
For the left and right sides, add 7/16 in. to the depth of the cabinet; for the bead molding on the front of the cabinet, add 7/8 in. to the cabinet’s front dimension (7/16 in. for each outside corner).
Finding the Right Angle…and the Ellipse
If you look hard enough, you can find hidden right triangles in places you never even imagined.
A vent pipe or circular chimney that passes through a roof or sloped ceiling is a perfect example.
| If you read “The Elegant Ellipse,” then you know that a cylinder or pipe that is cut (or intersected) at an angle creates an elliptical shape, and that shape is defined by a Major and Minor axis. |  |
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The Minor axis is simply the diameter of the cylinder and doesn’t change, but the size of the Major axis changes based on the angle (or pitch) of the intersection. |
To find the length of the Major axis:
| Enter the cylinder’s diameter as RUN. Enter the roof slope (inches of rise per 12 in. run) as PITCH. Solve for the DIAGONAL. When using the rise/run ratio of a roof, remember to hit the “Inch” key when entering PITCH. |
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With the Major and Minor Axes determined, the string method can be used to trace out the required shape. This method obviously won’t be used often in rough framing, but it’s a helpful trick to know when the cut has to be finish quality!
For more details on construction calculators and construction calculator mobile apps (convenient for the jobsite!), check out Calculated Industries’ Construction Master Pro, the mobile versions of Calculated Industries’ Construction Master Pro, and BuildCalc.
(SketchUp drawings by Wm. Todd Murdock; this article originally appeared on GaryMKatz.com)
If you need larger numbers, any multiple of 3, 4 and 5 will work. Like 6,8,10 or 9,12,15, or 12, 16, 20
Dan,
You betcha you can, and that’s the way the carpenter on my dad’s jobs used to do it, too. He had one of those tables on a piece of paper folded up in his wallet. But we don’t have to do that anymore! That’s the point of the article. You can square up anything so much faster and so much more precisely by finding THE EXACT DIAGONAL measurement.
Gary
Gary I’m struggling with finding back angles to say 6/12 or 8/12 roofs ” when a roof runs into another at opposite angles. I’m trying to figure out how we get the number. Besides just memorizing… A couple guys I work with have their own way but it doesn’t make sense. They have come up with their own way of visualizing it and that’s what I’m missing.
Thanks
Lewis
Lewis,
Boy am I the wrong guy to ask! I have a serious learning disability when it comes to math and angles–even arithmetic is touch on me. I can follow simple geometry…if someone is patient enough to explain it slowly and repeatedly. So I hope Mike Sloggatt or Todd Murdock or maybe the master–Sim Ayers–will answer your question.
Gary
Lewis,
What application are you specifically looking for–shed/porch roof tying back onto the 6/12 or 8/12 roof? Or am I missing the boat here?
Ed
brilliant! i love it. wish i had payed more attention to sine cosine and tangent in high school.
Great article. Thanks Gary!
And there is a^ + b^ = c^ = d^
H/V Rafter Lengths
Richard,
I’m not familiar with that one at all. But I am very familiar with this one: A3 + B3 = Z3
Gary
For the diagonal of square/(rectangles), same as Common lengths.
a^ + b^ = c^ or where a = 1, and b = 1, 1^ + 1^ = 2^, c (Hypotenuse) = ^2
For the diagonal of cube/(cuboids), same as H/V Lengths.
a^ + b^ + c^ = d^ or where a = 1, b = 1, and c = 1, 1^ + 1^ + 1^ = 3^, d (Hypotenuse) = ^3
Richard,
That’s not what I was referring to at all, though maybe I was being as obscure as you are. I was thinking of Fermat’s Last Theorem (which has nothing to do with roofs, but a lot to do with tattoos).
Gary
Unlike Fermat’s Last Theorem (or tattoos ?), which can take a long time to explain, . . . . just a footnote to the third dimension jotted down in the margin . . . , nothing more. (Now I know my AB . . . Z’s)
2^ = 2 Squared
^2 = Square root of 2
I had a loud, military, math teacher who told us “I will now teach you trigonometry in 5 seconds.” Then he shouted, “Sine OH, Cos AH, Tan OA.” We all got it in less than 5 seconds and it’s stuck ever since.
I learned it as SOH CAH TOA about 60 years ago.
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent
I don’t get it…. :)
This conversation brings back memories from my apprentice school days: Sin/Cos/Tan::Oscar Had/A Heap/Of Apples
Can never forget it!
Great article. Modern electronics have made many tasks easier, faster and more accurate.
I was wondering if there is an iPhone app that will do everything that the Construction Master Pro calculator pictured will do. A smartphone app would mean one less gadget to keep up with.
A lot of carpenters are using BuildCalc on their smart phones. It’s a pretty awesome tool. Because a smart phone is really a computer, it can organize and display information that a calculator can not display. You’ll see a lot of screen shots taken from BuildCalc in TiC articles.
Gary
There is a Construction Master app already and many others. I love mine and didn’t even know about how to use the rise, run and diagonal features.
http://www.calculated.com/
Thanks for the great explanation and examples!
I learned it as SOH CAH TOA about 60 years ago.
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent
Oh…now I get it. I’m very very slow with math.
First time I had ever even heard of “3,4,5” method was about 5 years after I started in the trades. I was taught the Pythagorean Theorem in middle school and just always used it. Still prefer to use that instead of the 3,4,5 method.
I have the construction calculator app for my phone, but find it much easier to just get specific apps like “square calc” that does it without having to remember all the super secret key strokes to get the answer on the calc. app.
So what happens when the batteries die?
Oh my! What happens when the batteries die on your PHONE?? You gotta go home!
There is way too much reliance on electronic devices in the trades!
I use my calculator everyday on the job, but I also have been trained in the fundamentals of geometry, something desperately lacking in the carpentry field!
If my batteries died I would resort to the same skills builders have used for thousands of years codified by Euclid in his Elements, or just pull out my framing square (another thing sorely lacking in the field, a carpenter who actually knows how to use a square).
That said, excellent article on the C.M. and it’s manny uses, Kudos Mr Katz!
If your batteries die, your truck won’t start, your rechargeable tools won’t work, and you’ll be deaf because your hearing aid won’t work. Like it or not, the gadgets are here to stay.
Who said anything about not liking gadgets?
It’s the heavy reliance on gadgets without a grasp on building fundamentals that’s problematic.
I was working on an industrial project building and setting forms, the layout party chief lost his CM and was not able to do basic geometrical calculations and decimal conversions, holding up the the form gang. One of the old timers steped up with a pencil and a scrap piece of wood, getting us back on track, not only was he able to figure the geometry, he could fire off 100’ths to fractions in the time it takes to press the on button.
TIC posts great articles like the this one, and gadgets like the CM and smartphone are very powerful tools, but lets not forget that the most powerful tool a carpenter owns: his brain, if you don’t use your tools they get rusty!
While 3,4,5 is the most common right triangle, there are an INFINITE number of others that are not multiples of 3,4,5.
The first few “Pythagorean triples”:
( 3, 4, 5 ) ( 5, 12, 13) ( 8, 15, 17) ( 7, 24, 25)
(20, 21, 29) (12, 35, 37) ( 9, 40, 41) (28, 45, 53)
(11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)
(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)
Yeah yeah yeah…but a construction calculator gives you the EXACT length of the hypotenuse to within 1/16″ or even better, so you can measure right from your corner stakes and not have to pin stakes in at arbitrary locations. What a time saver. And far far far more accurate.
Gary
Yeah, Pythagorean Theorem! Something totally lost on me after high school until I starting building. Great explanation Gary and Mike.
Thank the Greeks for all the construction geometry we use every day to build accurate stuff. Also, any one know where to get a good greek Gyro?? :-)…
Sarantos
NJ relocated to SC.
A great free site with tons of different construction calculators is
http://www.blocklayer.com/Default.aspx
DeWalt has a App but only for the iPhone and iPad for now, http://dewalt.cengage.com/mobilepro/
Thanks for the continuing ed. !
Math is like baseball learn your fundamentals, practice and apply.
Pythagorean Theorem – and its variations solves most building squaring, rafter lengths in our trade.Rafter books with tables,
Framing squares with brace scales, rafter lengths, hundredths
scale were older forms of a calculator. I learned old fashioned way by
memorizing multiplication tables, long division over and over
pencil and paper. As calculators came out and improved over the years we must embrace the technology,anyone younger is how they live and breathe. Learn your vocabulary read the directions
that comes with the calculator , practice , practice, practice.
Use your neumonic- to remember such as oscar had a heap of apples etc. sine, cosine, tangent .
Great forum !!
The more we know , the easier to to choose an efficient calculation method to apply.
Hello! Since you seem like the triangle expert, is it possible to cut crown molding along the long side of the attached window (crown goes to the ceiling), then miter another crown molding piece on the vertical side? Please advise. Much appreciated!
You can actually use Pythagorean for much more if you get the hang of it, even 3D distances are a breeze, especially if you use a calculator. I’m not sure of etiquette here but you can check out a nice explanation at “measure-any-distance-with-the-pythagorean-theorem” at Better Explained
Hope this is not a dumb question . I am really trying to learn. Under the heading “Finding Right Angles in Framing” using the bay pop-out illustration, that the diagrams do not show studs at 16″ on center? Would that not be a problem when covering the pop-out using standard 4’x8′ sheets of plywood or OSB?
I am laying a ceramic floor, and the architect has a spec of a 30 degree layout. Is there a formula or general idea to assure the 30 degree is correct….for instance 90 degree is a 6,8,10
Help I am a 45 degree master but this 30 degree has me stumped!
Seth, the rise of a 30-degree triangle is half the length of the diagonal (the diagonal is also called the hypotenuse). The run is not quite such a nice fraction, it is approximately 0.866 the length of the diagonal, or if you prefer 1.732 times the rise. Hope that helps.
Another application for your cabinet crown example would be trimming in the attic scuttle. You would subtract from your opening how much wood you want the drywall to sit on (usually 1/4″ each side). Then make your cut mark on the short side of your casing. Beginners stuff but I’ve seen a few guys struggle with figuring out the cuts on those.
I’m try to figure an easy way to figure out angle cots for windows. I make solar screens for the exterior of windows and figuring out the cut at the top using a 45 degree cut at the bottom. The bottom is square. Top has an angle like a trapazoid.
Okay, Forgive me but I am absolutely new to woodworking, measuring, etc. So lets say one is building a simple sawhorse. The legs are sticking out at an angle, how does one figure out what angle to cut the bottom of the legs so that they sit flat on the floor? Tightrope69@gmail.com
I’m TERRIBLE with angles! So if I were you, I mean, if you were me, you’d build those horses, then take a piece of 1×2 about 12″ long (or 1×3 or 1×4, whatever is tall enough to cover the high-point of the sawhorse leg), set that piece of 1x on the floor alongside each leg, then trace a line across the top of the 1x onto the sawhorse leg–like scribing the leg to the floor. Then use your circular saw to make the cut. No math. :)
Just finished reading this article and I have to say, I’ve been struggling to find the right angle for some of my projects. Thanks for the tips on using a combination square and a speed square, they definitely clarified things for me. Going to give it a try on my next project and see how it goes!
I completely agree with your point about taking the time to find the right angle. It’s amazing how much of a difference it can make in the quality of the final product. I’ve been there before where I rush through the process and end up with a wobbly or uneven piece. This post is a great reminder to take my time and get it right.