Best Way To Verify A 45 Degree Angle? - Framing


Refresh my memory,
We are just starting a new frame, and I can't seem to remember how to mathematically verify the 45 degree angle on this foundation wall.
Normally the other (parallel) walls are long enough on a house, that I can run a 90 degree angle and get it that way.
So this is what I have.
The yellow lines are established as square.
The red circle is a 45 degree.
What I want to know, is
can I deduce the length of the red line given the length of the black lines, and the angle of the red circle?
For instance, If I pull the tape 24' on each black line, I get 44' 4"1/8 as the red line measurement.
BUT I cheated, I used sketchup. Before Sketchup as invented, I would have run string lines and tried to get parallel measurements fron the string.
Anybody know the proper formula for this?
BTW what A nice view on this Acreage!

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Are the lengths of the black lines the same? A line that bisects the point in the circle will create two equal right triangles. From there, Pythagoras takes over.

I can make the black lines what ever I want them to be.
I know what you mean, But I need to VERIFY the 45 degree corner is actually 45 degrees. Which means I need to be able to find out what the length of the red line based on what the angle is and my given (black lines) does that make sense?
I think your 45 is off by a little if I'm using the calculator right. 0.7deg off I think

If i am not mistaken you could use sine or cosine law to figure it out. Or if possible, create a 90* and bisect it.
I am quite curious to find out the solution to this!

Normally I would do the 90 and bisect, but there isn't any walls long enough to do it.

It's an isosceles triangle so;
Base angle sin (180° - 45° = 135°/2 = 67.5°) =0.92388
0.92388 x 24’ = 22' 2 1/16” (base of the right triangle of 1/2 of the isosceles triangle) x 2 = 44' 4 1/8"
The rise is 9' 2 1/4".

Half of your 44' 2 1/8" line is 22' 2 1/16". 2- 22 1/2 deg right triangles form a 45. So diagonal is 13" per foot. (13"/ft) X 22' 2 1/16 = 24' 1/4". You're off like a 16th in 22 feet.

Or what he said

You could double check the 45° angle by marking the center of a pulled string across the base, then check the rise (9' 2 1/4" assumes your 2 legs are 24').

Yeah, after I read your last post, I was planning to do that first thing in the morning.
* edit, thanks guys I learned something valuable tonight*

Easiest way is a CM with the trig function or the BuildCalc app it has trig built into it.

Yeah the isosceles triangle Whenever I see a triangle I always try and figure out a way to make a right triangle. Then 345
But it seems that they already beat to helping you.

If you need to be 100% sure! set up batter boards around the foundation and extend the closer yellow line to the right with an actual string or laser.
Then you'll have a true reference line to pull from to check that upper right corner.

When you normally bisect them how do you go about doing so? I guess on a wood subfloor its easy to strike arcs and locate intersection points.. but on a foundation with nothing there... How

Foundation layout relies on math. The excavator or the person placing the footings has nothing to measure from at first. Just to square up the the batter boards to dig the hole you need to us math. Once one corner is set you pull a length for the first wall, set the batter board with the over dig, using math you find the length of the diagonal to set the second corner square.
All the roofs you cut are nothing but isosceles triangles you turned into right triangles.
You gotta trust the math.

I would have probably done that Friday, but the wind was at 30 miles an hour, so I wouldn't have trusted the string anyway
I wasn't stumped with this anyway, I just wanted to remember the math for it.

How close were you?

How close was the concrete?

I don't know, that's why I asked you.

hehe, were you asking me how close the concrete was to 45 degree?
I don't know, I'll know monday morning.
I'm kind of scared to find out. The 2 long walls are out by an inch over 30'

Did you actually pull that 44' 4 1/8" or is that what you got with SketchUp? If you actually pulled that measurement, you need to find some new concrete guys. They're setting the bar pretty high on quality!!!!

I'm willing to bet it will be over an inch out. At least. Sometimes these guys are an inch out over 10' I had to use a triple rim board on an entry way once.

Didn't read all the other posts so sorry if its already been said.
Knowing the lengths of the 3 sides you can use the law of cosines.
The C angle is across from the c side, using this for all sides you can determine all angles. Or you can find the one angle and then use the easier law of sines.

Law of cosines -
Law of sines -

The only knowns we're the lenght of the legs and one angle.
The base length was unknown. That is what he was looking to determine mathematically.

For the heck of it, try solving using square roots.

It's ok... We're working off of a foundation that'd out 3 inches in 20 feet..........
It's an addition. HO said go with it. Alright, boss.....

Verify that the back wall and side wall are square to each other first.
Measure length of side wall to corner. Measure length from back wall to long point of angle wall. Subtract to find the difference. Use this number as the lengths of a right triangle and calculate the diagonal. If your angle wall is a 45, the number won't lie.
Example: Side wall is 30 feet. Length from back wall to point is 40 feet.
This leaves 10', or 120". Diagonal of 10' is 169.705".
If your diagonal doesn't match this length, the angle is off.
The same can be accomplished with unsquare reference walls by snapping square lines on top of the walls.
As for the 45 wall, you only need two numbers to establish a 45. Never trust the mason.

Using this "equal right triangles" school of thought, w/a 135* deck angle, and a CMC.
Set Pitch = (180 -135) = 45* / 2 = 22.5*
Enter 22.5 [Pitch], 10' [Run], [Diag] = 10' 9-7/8"
The Black lines are 10'9-7/8" each,
line is (2 x 10') = 20'

What's the yellow line length?
That will b the diagonal of half the square of the two blue lines that should be equal.
Once you have the yellow line length from that rectangle you can get the exact length of the two blue lines.
What are the rectangle measurements?

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How can this be if the OP states in the first post that the black lines are 24'? It has also been determined that the red line is 44'-4 1/8"

I mentioned in an earlier post to use a calculator to find the sine. If you have an iPhone with iOS 7, you can use the calculator in Control Center (swipe up from bottom to access), turn the phone to landscape you have a scientific calculator with the trig keys. My guess is Android has something similar.

If the black line is 24', then the width of the rectangle should be 33' 11-5/16".
Just view the yellow rectangle in plan view with a 45° triangle off the end.
The width of the triangle is the diagonal of the 24' black line.
24 [Feet] [Run]
45 [Pitch] [Diag] Returns -33' 11-5/16"

Or give the dimensions of the rectangle and the will give the dimensions of what the black lines of the triangle should be.

The reason I didn't us the "blue lines" to help is, I don't know if one or both or none of those walls are correct either. (not yet) So using that a benchmark to verify anything would likely not work out.

What I'm saying is YOU make it work. You have the length and width of the rectangle yellow lines I drew. Those should be exact. If the foundations not square, you make your lines square. Now you can make the blue lines work out to an exact 45°.
What are the dimensions of the yellow rectangle?

This is realistically the way I am going to verify this and other 45 degree corners from now on.
It's simple and easy for me personally to understand.

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Not sure, the plans are on site. And I don't remember,
I re-read what you said.... you are right I agree

Whatever way is the easiest for you to undrestand is the best way.
I'm just suggesting to you that using the width of your rectangle is all you need to create the two black lines of your triangle, or the blue lines I made.
Since your black lines are 24', the width of the yellow lines in the rectangle should be 33' 11-5/16".

Here's a drawing using the rectangle.

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In his follow-up 2nd post he said he could make the Black lines anything he wanted. I posted an approach variation using shorter dims. Still forms a perfect 135* deck angle.
The concept is the same.

That believe that would be the slope gain, or secant value of the slope angle approach which would use the secant of 22.5deg. = 1.0823922. The secant is = to the hyp/the adjacent side.

Since the hyp of your triangle is 24 ft, you would divide the hyp(24') by the secant to get the opposite side of the right triangle.
since the secant of 22.5deg = 1/cos of 22.5,

24/(1/cos 22.5) = 22.1731 ft
since there are two legs in your diag...
22.1731*2 = 44.3462 ft
.3462*12= 4.15", or 44ft 4.15" ...a little over 44ft 4 1/8"

I use the secant method on my rail spread sheets for layout level to slope conversions.


For marking the foundation and getting the exact measurements down and the correct 45° angle...all you need to do once you have the rectangle marks and lines square is pull off the corners of right side of the drawing wall 24' until the two 24' lines bisect each other. That sets your 45° mark and you snap your 24' lines from there.

I just wanted to re-iterate, this did NOT have me stumped.
This is a plan view of the house. (I had some plans at home that I quoted from)
We only had about 45 minutes on Friday to work on the layout. So we spent maybe 3 minutes scratching our heads.
The picture probably show a bit more clearly. Baring measuring off of a suspended string line,in the middle of nowhere. I do not see a way that I could have gotten the 45 on site without using the math we were discussing. Like I said, the walls are not friendly to making square corners.
I would have gotten it anyway, but I was interested in the way to mathematically get the answer. If for no reason, than to verify my work.
working through this thread helped me remember things I used to know, and learn a few new ways too.

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How come no one has brought up the 9.362 factor....


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