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.|
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.
|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.|
|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.
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.