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Finding the Right Angle

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.

fig 1-1

(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.

fig 2-1

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.

fig 3-1

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

fig 4-1Laying 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.

fig 5-1

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.

fig 6-1

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. fig 7-1
fig 8-1 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!). fig 9-1

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!

fig 10-1

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. fig 11-1

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!

fig 12-1

Finding Right Angles in Finish Work: Cabinet Crown

Foundations and framing aren’t the only places where right angles occur.

cabinetphoto-1I 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?!”

 fig 13-1

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.

fig 14-1

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.

fig 15-1

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. fig 16-1

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).

fig 17-1

Finding the Right Angle…and the Ellipse

If you look hard enough, you can find hidden right triangles in places you never even imagined.

fig 18-1

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. fig 19-1
fig 20-1 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:

fig 21-1

 

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.
fig 22-1

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)

Comments/Discussion

31 Responses to “Finding the Right Angle”

  1. Dan Miller

    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

    Reply
    • Gary Katz

      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

      Reply
  2. ben

    brilliant! i love it. wish i had payed more attention to sine cosine and tangent in high school.

    Reply
    • Gary Katz

      Richard,
      I’m not familiar with that one at all. But I am very familiar with this one: A3 + B3 = Z3
      Gary

      Reply
      • Richard Birch

        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

        Reply
        • Gary Katz

          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

          Reply
          • Richard Birch

            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

  3. Terry Mickelson

    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.

    Reply
    • R Hutchins

      I learned it as SOH CAH TOA about 60 years ago.
      SOH: Sine = Opposite / Hypotenuse
      CAH: Cosine = Adjacent / Hypotenuse
      TOA: Tangent = Opposite / Adjacent

      Reply
      • Randy

        This conversation brings back memories from my apprentice school days: Sin/Cos/Tan::Oscar Had/A Heap/Of Apples

        Can never forget it!

        Reply
  4. R Hutchins

    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.

    Reply
    • Gary Katz

      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

      Reply
    • D Craig

      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!

      Reply
  5. R Hutchins

    I learned it as SOH CAH TOA about 60 years ago.
    SOH: Sine = Opposite / Hypotenuse
    CAH: Cosine = Adjacent / Hypotenuse
    TOA: Tangent = Opposite / Adjacent

    Reply
  6. brew

    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.

    Reply
    • Gary Katz

      Oh my! What happens when the batteries die on your PHONE?? You gotta go home!

      Reply
      • Jay Stein

        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!

        Reply
        • Dave O

          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.

          Reply
          • Jay Stein

            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!

  7. Dave

    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)

    Reply
    • Gary Katz

      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

      Reply
  8. Andrew

    Yeah, Pythagorean Theorem! Something totally lost on me after high school until I starting building. Great explanation Gary and Mike.

    Reply
  9. Sarantos

    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.

    Reply
  10. Sean Mahoney

    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.

    Reply
  11. Gary

    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!

    Reply

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