I just had this young guy ask me this question. If a 12/12 = 45 degrees then why doesn’t 6/12= 22.5 degrees?
Since I didn’t know I gave him my standard answer to such questions “It’s code”.
Please help me , the kid looks up to me and I would hate to let him down. 🙂
Dave
Listeners write in about removing masonry chimneys and ask about blocked ridge vents, deal-breakers with fixer-uppers, and flashing ledgers that are spaced from the wall.
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Replies
Draw the gable end of the roof as a triangle. Drop a vertical line from the apex of the triangle straight down perpendicular to the base.
You now have two right triangles. Look at only one of them. The two non-hypotenuse sides of the triangle (legs) are equal length, both 12, so that means the angles are 90-45-45.
Do the same exercise for a 6/12 pitch. YOu have a 30-60-90 triangle, don't you? I think.....
Tell the fella it's trigonometry. Non-intuitive.
Edit: YYO! I AM WRONG! Where is PI?
It's not a 30-60-90, and the angle is something like 27. Sheesh.
"The blue and the dim and the dark cloths
Of night and light and the half light..."--???
Edited 11/15/2002 6:34:31 PM ET by Theodora
Edited 11/15/2002 6:39:53 PM ET by Theodora
Dave: Take the rise...6...divide it by the run...12....this is .5
The tangent of .5 is 26.56 degrees.
Take the rise 12.... the run ....12......divide the rise by the run..and you get 1
Take the tangent of 1 and you get 45 degrees.
Ah, yes, perhaps it's time to tell the tale of the Fair Indian Maiden Princess Sohcahtoa. From the Jee-o-metree Tribe. She traveled with Lewis & Clark and showed them how to estimate the vertical height of topographical features such as waterfalls and canyons.
That Sacagawea gal did hardly any work at all, yet was given all the credit.
sohcahtoa? never heard of her - but we were told in great detail about the volcano Toacahsoh when I was in school.
They told you about Toacahsoh? Without mentioning Sohcahtoa?
I'll fill you in...Sohcahtoa estimated the height of Toacahsoh for Edmund Hillary. Not many people know that he used Toacahsoh as a prep climb for Everest.
Some Old Hippie Caught Another Hippie Tripping Over Apples.Steelkilt Lives!
Looks like someone has way too much time on their hands.
With SohCahToa the princess I hope we aren't going in the direction of the Fugawee indian tribe! :)
After reading this thread I ask myself "Why didn't they ever consider math a foreign language?"Tamara
All of these are nice answers to the math problem, but none of them answer the fundamental nature of the young man's question... That is, WHY is a 6/12 slope not half the angle of a 12/12 slope (or conversely, a 24/12 slope is not twice the angle of a 12/12 slope)?
OR: Why shouldn't it be? (Because it seems like it should be).
The answer is more philosophical than anything else. It comes down to this: because that's the way God designed the universe.
It's akin to asking, "why is the ratio of the radius of a circle to it's circumference 3.1415..." Instead of, say, 1.62?
But speaking of angles and the ratios of the sides of a right triangle (the pitch of a roof, in this case) is fundamentally the same sort of question. The technical answer to the question (why is a 6/12 pitch not half the angle of a 12/12 pitch?) is this:
SLOPES AND ANGLES ARE TWO COMPLETELY DIFFERENT ANIMALS.
Because while the range of possible angles between 0 and 90 degrees is a FINITE set, the slope (the pitch) of the hypotenuse of the angles is an INFINITE set. A 0/12 roof is a horizontal line with a slope of 0. But the pitch of a vertical (90 deg. angle) roof is infinite. Half of infinity is still infinity, and twice zero is still zero. The young man was asking the wrong question.
He, and we, are making a moral judgement about a physical observation of the cosmos. It just is what it is. I know, I know, but WHY is it?
SunToad,
Geez, that was a mouthful Suntoad. Actually, the range of possible angles between 0º and 90º is not a finite set as you mentioned. It also is an infinite set. As a matter of fact, the range of possible angles between 0º and 1º is an infinite set also, (not that it has anything to do with the original question).
The answer also has nothing to do with philosophy, or for that matter, the way God designed the Universe.
It also has nothing to do with making moral judgements, or physical observations of the cosmos. ( got anymore of that good stuff?)
It's simple trigonometry. The graph of the tangent function is not a straight line. Therefore, if you plot the tangent of say, 45º, which is = 1, or 12/12 , it would be incorrect to assume that the tangent of 22.5º = 1/2, or 6/12 . If that type of reasoning were correct, then the tangent function would have to be a linear function.
I did enjoy your post. And yes, twice zero, is still zero. I'm with you there.
Ken"Nothing's so hard but search will find it out"--Robert Herrick
Hi Ken,
I know. But you’re splitting hairs and missing the point. While the set of angles for any real slope between horizontal and vertical ranges between 0-90 degrees, the slope itself will range between 0 and infinity. Remember, the tangential function is merely a fancy term for the slope/pitch of the triangle’s hypotenuse (in this case, the roof).
I understand what the tangent function looks like graphically. But that was not the question. It was WHY?
And your answer begs the question. You are answering the riddle of "why isn't half the slope of a 45 degree angle half of 45 degrees" with an observation of fact: "because the tangent is 26.5 degrees". I know. But why? Intuitively, the carpenter says it ought to be 22.5. And OUGHT is an ethical question.
Anyway, in dimensional analysis, one cannot mix units (of measurement). Angles of a right triangle are a function of the degrees of a circle--units are degrees or radians. Slope is a function of ratio--no units. A 12/12 pitch is only twice a 6/12 pitch in a comparative sense. But not in a quantitative sense. It only seems like it should be.
See, it's philisophical. Not elementary, my dear Watson.
Sun Toad --
I'm too lazy to go find a book right now, but aren't radians also unitless?
ragnar,
That's correct. Radians are dimensionless numbers.
SunToad,
If one of the legs of a right triangle is held constant, and the length of the other leg is doubled, the hypotenuse will change also. Therein lies the problem.
In the new triangle, the right angle still has 90º, but the other 2 angles will change. One angle is formed by the hypotenuse and the constant side, the other by the hypotenuse and the doubled side. It is the new ratios between the length of the hypotenuse, and these legs, that determine the new measures of the angles.
You've got to look at the whole picture.
Got to take a break. I'm going to gather up all the philosophy that I can get my hands on, plus $1.51, and see if it gets me a tall coffee at StarBucks <G>
"That's correct. Radians are dimensionless numbers."
Not really, but you are mis using the term. Radians are the dimesion.
The MEASUREMENT is of an angle. That measurement can be expressed in the units of radians, in degrees, in minutes, in seconds, or in quandrants.
Just as in the measure of liner distance can be expressed in units of feet, meters, miles, fathoms, or furlongs.
Okay, okay, we obviously have a mathematically astute audience here. So, here goes: Will someone please explain, in simple and brief terms, how to trisect an angle? Can't be too hard, you all have proven your skills at bisection, and we're just adding one other, small, insignificant line...
Regards,
Rework
From what I recall in high school geometry, there is no elegant graphical solution to trisecting angles.
The typical approach to bisecting an angle hinges on bisecting a line. However, trisecting a line will NOT yield a trisected angle. Go figure. <g> You need to trisect an arc.
There is a way to trisect an angle, but it's a bit trial-and-error. Start with your angle. Sweep an arc between the two sides of the angle using the angle vertex as your center point. Now, use your compass to "step off" the arc in exactly three equal segments (this is the part that takes a little trial and error). Once the arc has been trisected, simply draw lines from the angle vertex to the trisected points on the arc.
Ragnar
Edited 11/17/2002 4:13:07 PM ET by ragnar
Edited 11/17/2002 7:44:34 PM ET by ragnar
Y’all just like to argue.
Bill is right, though. Radians ARE the dimension—the unit of measurement.
The dimensions of a ratio are unity. No units. A ratio is a comparison. That’s why all of the trig functions have no units.
Angles, on the other hand, are measured, and any measurement must have units, else it’s a nonsense statement.
SO. The logic of asking why a 6/12 pitch is not half the angle of a 12/12 pitch is fallacious. Apples and oranges. One is a ratio, one is a measurement.
Otherwise, at what pitch then does a roof become a wall?
"Otherwise, at what pitch then does a roof become a wall? "
When you put a dormer in it ;-PSteelkilt Lives!
"All of these are nice answers to the math problem, but none of them answer the fundamental nature of the young man's question.."
Actually, I thought I answered it pretty well back in reply #13:
"When you take a 45° angle and divide it in half, you're dividing an arc in half. When you deal with roof pitches, you're dealing with right triangles with different lengths on the vertical leg."
I was hoping to get some feedback on that explanation. Did it make sense? Mom's Travel Agency - ask about our guilt trips.
As you have stated them this is the cotagent, i.e., the cotagent (or angle whose tangent is;) of a 12/12 = 1 pitch is 45 degrees. Conversely the tangent of 45 degrees is 1.0
At 45º tangent and co tangent are equal, to 1 that is!
Pi you learning any thing here?
Your Math students might benefit from all the wisdom here.
If nothing else you can use this as a bad example!Do not try this at home!
I am a trained professional!
I do admire someone who will correct this kind of thing and leave the original showing. I'd have deleted that sucker. :)
Aha, but you may not have noticed, since I don't try to deliberately flaunt it, but.....
I am not a guy...
It is OK for me to look less than omniscient occasionally, or ask for directions!
(hee hee hee)"The blue and the dim and the dark clothsOf night and light and the half light..."--???
Okay, might as well add my version in as well. Goes way back to junior high school, and I've never forgotten it.
Oscar Had A Hold On Arthur
Sin Cos Tan
the ratios you're talking about would mathmatically be called the "tangent" of an angle, ie the tangent of 45 degrees is 1 (=12/12). If you made a graph of all the tangents of all the angles (an infinite number of angles, therefor a smooth line), that line is not a straight line, it's a complicated curve - ie the tangent of 45 degrees is not simply twice the tangent of 22.5 degrees. A lousy anwer I know, but that's why I'm not a math teacher.
Or you could just say, "go ask your mother".
In case you don't keep trigonometric tables in your pocket while out on a job site, remember that your trusty speed square will give you a general conversion of rise/run numbers to degrees.
Dave,
View Image
Edited 11/16/2002 12:42:38 PM ET by J Fusco
I knew Joe would chime in on this - he seems to know the un-intuitive, well, intuitively.
Since no one ever expounded upon the mnemonic; but only tried to look clever (myself included)
SOH
Sine of an angle = Opposite over Hypotenuse
CAH
Cosine of an Angle = Adjacent over Hypotenuse
TOA
Tangent of an Angle = Opposite over Adjacent
If you don't have a calc with trig functions, use the tables.
http://www.math2.org/math/trig/tables.htm
Rise and run describe a ratio, the only way that "half of 45 is 22.5" is when you are describing arcs within the same circle.
Picture that you are drawing the right triangle within a quadrant of a circle (for example, between the 12 and the 3 on the face of a clock), the hypotenuse is the radius of that circle. In a 12:12 pitch, the hyp. is pointed midway between 1 and 2 on the clock. Draw a line straight down from that (the rise) and you will notice the run makes it about 70% of the way from the center to the outer diameter to meet that rise.
Now picture the same clock, same situation, same RUN, but this time a different pitch. (let's use a 6:12). You will notice that the RISE of 6 does not reach the outer diameter of the old circle. So, in truth, 6:12 is not wholly equal to half of 12:12.
Then again, using a speed square works, too.Steelkilt Lives!
I deal with this question a lot regarding trusses, so I'll throw in a thought or 2 here.
Start out with a 1/12 angle, which is 4.8°. This triangle is 12" across the base, and 1" on the vertical leg.
Take a 2/12 angle, which is 9.5°. This triangle is 12" across the base, and 2" on the vertical leg. Subtract the 1/12 angle of 4.8° and you'll notice that this angle has only increased by 4.7°.
Then look at a 3/12 angle, or 14.0°. That is only an increase of 4.5° from the 2/12 angle. (The "law of diminishing returns", I guess.)
Take the diference between a 12/12 angle (45°) and 11/12 (42.5°) and you'll see the difference is only 2.5°.
When you take a 45° angle and divide it in half, you're dividing an arc in half. When you deal with roof pitches, you're dealing with right triangles with different lengths on the vertical leg. (The horizontal leg is always 12")
-------------
This question comes up with regards to trusses every once in a while. There's a rule of thumb that architects like to use that "any scissor truss will work, as long as you have a 2/12 difference between the top chord and bottom chord". But that's not always true.
Take a scissor truss that's 4/12 (18.4°) on the top chord, and 2/12 (9.5°)on the bottom chord. The relative angle between the top and bottom chords is 8.9°.
But change the pitches to 8/12 (33.7°) over 6/12 (26.6°) and the relative angle is only 7.1°. The truss may not work, or may have too much horizontal deflection.
That oughta either answer your question, or bore you to tears.
Why do they call it a TV set when you only get one?
Dave,
Tell them that a 6/12 is not 22.5º for the same reason a 24/12 is not 90º
That will get them thinkin'
T
Do not try this at home!
I am a trained professional!
If you truly want to answer this man that looks up to you in the best way, tell him that idiots called mankind has not come up with a math system that is universal and encompasses every part of life and both pitch and angles are exactly that! Different measurement systems because "idiot mankind" are too busy arguing about which is right when none are truly! Then show him these post as proof! Good luck and I truly hope you inspire this young kid in a good way!
Ok
I want a chance to answer this 20 years too late. I had the the same question but the way I am thinking to answer is: the curve of a circle vs. a rectangle. a pitch at 12/12 is 45° but if you use that hypotenuse to draw a circle with the hypotenuse as a radius from the center then draw a pitch 0f 6/12 from the same center point and the hypotenuse of that triangle creates a smaller circle and if you follow that line to the larger circle it doesn't divide it in half, since the leg of the triangle is not curved. I added a simple drawing but not sure if it will show up.
The main thing I'm pointing out is that the degrees are on a circle and pitch is a measured on a square or rectangle since the legs of a triangle are straight lines they won't match the curve of the circle
trying to add attachment again
Doug
https://www.mathsisfun.com/sine-cosine-tangent.html
Rise over run is the same as opposite over adjacent (both assume a right triangle)
Opposite over adjacent = Tangent of the angle.
12/12=1=Tangent of 45 degrees.
to get the angle directly from the tangent, the arctangent function was invented.
arctan(rise/run)= angle.
https://www.rapidtables.com/calc/math/Arctan_Calculator.html
Understanding some trig is a very useful skill when building.