Any Math Wizards - Figuring Pier Diameter to Encompass a 20x20 Square

[danieldd]

Confused yet? Well, so am I. Its been too long since I've been out of college.. and I wasn't a math wiz anyway...

So, I'm putting up a Pergola on our concrete patio. Been working on this for the past 2 years.. All the wood is cut and I need to get it out of my shop so I can do other things in the shop -

Anyway, I went to the city inspection department to pull a permit for the Pergola. Pergola dimensions are 10X12 foot with 9 foot height using four 6x6 inch posts located at each of the 4 corners.

I figured I could use the "Titan" post anchors to anchor the posts to the concrete patio, which I think are really cool. Talked to the inspector and he had no problem with me using them, except they are requiring me to cut a 20x20 inch hole with a depth of 12 inches in the concrete for each of the 4 posts. Essentially I am required make 4 concrete piers.

Once inspection is done, I will fill the holes with concrete AND then I can install my Pergola onto the piers.

Don't want to cut the concrete in a 20x20 square as the saw marks will overcut the concrete by a good 4 inches and look bad. Rather I would rather cut a circular hole, but it must be large enough to encompass a 20x20 area.

So, the analogy would be to fit a square peg (20x20) into a round hole. What would the diameter of the hole need to be to accomplish this?

 

[hansen1]

Well I won't claim that my math skills are the sharpest but based on what you are asking is you need to figure out what the distant from the center of the 20x20 square hole to any one of the 4 corners. To do this, I made a triangle from the center of the square to center of one side. This would make 2 of the legs of the triangle equal to half the distance of the square which would be 10".

So now what is the longest leg (Leg C) of a triangle with 2 legs (Leg A and Leg B) being 10" each. Based on what I recall, A squared + B squared = C squared. This would give you the radius of the circle. Just need to double to get the diameter.

So my prediction is you need a round hole with a diameter of 28.284".

Anyway care to check my math?

I made it more complicated than it needed to be. You could also do the same calculation by using each side of the square for the length of the Leg A and Leg B. Comes out the same.

 

[Steevo]

So, you want 400 square inches of surface? (20x 20)?

A circles area=3.14*(radius squared)

So, if you want 400 square inches of area, the circle needs to be approx. 22.6" across.

diameter=22.6
radius=11.3
3.14*(11.3 squared)=400.9466 sq in

 

[adcrawfo]

Approximately 22.5" in diameter to encompass the 20x20 area.


Sent from my iPhone using Tapatalk

 

[rburke65]

I think I'm missing something here. For a 10'x12" structure you need four 20"x20" holes in concrete?

 

[kindyr]

I got there in a slightly differant way than Hansen1, but my math came up with the same end answer.(I actually couldn't follow Hansen1's math)
My way was that the widest distance of a square is from one corner to the opposite corner. So you have a right angle triangle with the hypotenuse being the corner to corner measurement. the other two sides are each 20". so 400 + 400 = 800. The square root of 800 is 28.284, so the distance across a 20X20 square is 28.284" which would be the diamete needed for your circle.

 

[theoldwizard1]

My way was that the widest distance of a square is from one corner to the opposite corner. So you have a right angle triangle with the hypotenuse being the corner to corner measurement. the other two sides are each 20". so 400 + 400 = 800. The square root of 800 is 28.284, so the distance across a 20X20 square is 28.284" which would be the diamete needed for your circle.

Concur !

(I'm weird. I love math, especially Algebra)

 

[danieldd]

Okay..

thanks.

Question is now - who is right?

 

[danski0224]

I'd rather know who is in the avatar

 

[ed_h]

You want a circle with a diameter equal to the diagonal of a 20" x 20" square.

Diagonal of any square is (square root of 2) x (length of one side of square)

(square root of 2) = 1.414

Answer: 1.414 x 20" = 28.28"

 

[danieldd]

Got it!

So I've got to cut quite a large hole in the concrete! Geez!

I think the inspection department is asking for overkill on the piers. Unfortunately, if I want to abide by the requirement, I'm going to have to comply.

Still, that is a big friggin hole!

 

[srmofo]

I'm Getting 28.28427125 Inch Diameter

 

[kindyr]

I think both answers are correct depending on how the question is asked. If you are asking for the circular equivelant in area to a 20X20 square, then 22.5" diameter circle would have the same surface area. If you are looking for the diameter that will allow and actual 20X20 square to fit into it, I believe the 28.284" answer is correct.

my understanding is you are trying to determine how big a round hole would be needed to fit a square peg in, that would be to 28.284"

on a practical level it sounds like a lot of extra concrete than you actually need, so you may want to check with the building inspector regarding how large the post would need to be if it were circular.

 

[zkling]

Just to make sure I understand this correctly, you want to find that diameter of a hole that would allow a 20"x20" square to just barely fit in it? If that is the case you will need a circle with a diameter 22.28" to fix a 20"x20" square into it. This is commonly called circumscribed.

Now on the other hand if you want a circle that has the same area (in^2) as a 20"20" square. You will need a circle of diameter 22.57"

Finally if you want a circle that will fit inside a square hole that measures 20"x20" you would need a circle that is 20" in diameter. This one is the most obvious.

 

[breeaad]

Well I won't claim that my math skills are the sharpest but based on what you are asking is you need to figure out what the distant from the center of the 20x20 square hole to any one of the 4 corners. To do this, I made a triangle from the center of the square to center of one side. This would make 2 of the legs of the triangle equal to half the distance of the square which would be 10".

So now what is the longest leg (Leg C) of a triangle with 2 legs (Leg A and Leg B) being 10" each. Based on what I recall, A squared + B squared = C squared. This would give you the radius of the circle. Just need to double to get the diameter.

So my prediction is you need a round hole with a diameter of 28.284".

Anyway care to check my math?

I made it more complicated than it needed to be. You could also do the same calculation by using each side of the square for the length of the Leg A and Leg B. Comes out the same.

Hansen nailed it exactly, along with explanation.

 

[danieldd]

...... If you are looking for the diameter that will allow and actual 20X20 square to fit into it, I believe the 28.284" answer is correct.

my understanding is you are trying to determine how big a round hole would be needed to fit a square peg in, that would be to 28.284"

on a practical level it sounds like a lot of extra concrete than you actually need, so you may want to check with the building inspector regarding how large the post would need to be if it were circular.

Yes, you are correct

Also, it does appear to be a lot of concrete and I did specifically ask the inspector how big it needed to be if I were to cut a round hole. Essentially, he showed me the building code spec in the book and told me to make the hole large enough to meet the minimum specification: which is large enough to encompass a 20x20 inch square.

Personally, I think the size of the hole required for a pier is just too big, given this is only a 6x6 inch post. I can't argue the point with the inspector when the specification is right there in black and white.

 

[gorilla]

Can I ask how you will cut a 28.284" round hole in concrete? Can you get a core drill that big? I'm not trying to be a smart ***, just trying to learn something.

 

[1949 caddyman]

I bet the requirement is for wind. They want enough weight on the posts to keep them from becoming air born in a storm.

 

[B-Well]

so did you determine if you needed to fit the square in the circle or have square area = circle area?

sqrt of (side x side x 2)= diameter = 28 1/4

or

 

[LXCam]

Me thinks you guys may be over thinking the thought process. The city has on record a list of pre-approved pier footings for certian applications. Ask them if they have a standard for a circular pier. You'll most likely find they'll require a 24" diameter with a increased depth to between 12/15". The other method of compling with thier requirement and not having the over cut is to cut the perimeter to size, cross cut the diagonal, break some out and use a 4.5" angle grinder with a diamond wheel to clean up the inside if the corners. Then everybody is happy. GL

Oops I forgot something. Now if you find a diameter that is acceptable and you find a demo guy with the correct size core drill. You will still have to deal with the anchor he will have to drill and sink in order to hold his coring unit in place. So I have to ask what will be more unsightly to you, the potential of a slight over cut orthat anchor depending upon placement.

 

[sloppy]

Can I ask how you will cut a 28.284" round hole in concrete? Can you get a core drill that big? I'm not trying to be a smart ***, just trying to learn something.

Yes you can get a core drill that big and larger..

 

[danieldd]

so did you determine if you needed to fit the square in the circle or have square area = circle area?

sqrt of (side x side x 2)= diameter = 28 1/4

or

Your 1st picture is what I am attempting to accomplish. I'm going to get an estimate from the concrete guy today to have this done as I don't want to mess with buying a boat load of Quikcrete bags to fill these holes back up with. We're talking about a 1/2 yard of concrete!

 

[Chaz]

Why does the top of the hole have to be so big? Cant you cut a smaller hole in the concrete and excavate larger once below the slab?

 

[danieldd]

If I simply cut a sqare hole, then I can reduce the size to 20x20, but then I have to contend with the overcut sawcut marks in the concrete. I can avoid this by cutting a hole, but it has to be 28inches in diameter, to encompass the minimum 20x20inch requirement.

The city wants to insure I abide by the minimum requirement. They have to inspect the hole before I can go any further.

 

[Strouty]

If I simply cut a sqare hole, then I can reduce the size to 20x20, but then I have to contend with the overcut sawcut marks in the concrete. I can avoid this by cutting a hole, but it has to be 28inches in diameter, to encompass the minimum 20x20inch requirement.

The city wants to insure I abide by the minimum requirement. They have to inspect the hole before I can go any further.

Your CEO should be able to sign off on the smaller circle. All they need is to have x amount of concrete in the hole to support the beams. I would ask them again. Do the math for the amount of concrete. The circle you think you need is a lot more concrete and a waste of time, effort, and money.

The 20" by 20" by 12" deep square is 4800 cubic inches of concrete and the 28.24" diameter circle will be 7536 cubic inches. Basically double the amount of concrete. A short load from a concrete company will cost you an arm and a leg. You will probably have to pay for 4 yards at $100 a yard.

After looking at home many bags you will need, you may be better off paying for a short load. If you can get them to agree to the smaller circle with the same area as the 20 x20 square, it wouldn't be that bad and you could save yourself a few hundred bucks for an afternoon's work.

 

[CNGsaves]

So OP . . . gotta tell us what City, and what Country is forcing that massive sub-structure for a pergola??

Any pics of this "massive" pergola structure that will need 4 piers??

How deep do these core holes through concrete have to go??

 

[Jaguar Fan]

Maybe cut the square rather than a circular cylinder. Then to deal with the saw marks, tile the entire surface, or apply a coating over the concrete (like an epoxy, but one that is certified for outdoor UV exposure).

 

[reverendjonas]

What about a concrete chainsaw?
No over cut lines if you are careful.

 

[Garage Dog]

"The other method of complying with their requirement and not having the over cut is to cut the perimeter to size, cross cut the diagonal, break some out and use a 4.5" angle grinder with a diamond wheel to clean up the inside if the corners... GL"
-------------------------------------------------------------------------

I think LXcam has the right idea here.

Cut to the corners then clean them up with a grinder. Drilling a series of holes along the remaining corner cut lines with the right size masonry bit and use a chisel to clean it up would probably work as well.

Far less expensive for you or a mason to accomplish.

 

[Kevin54]

28.284 it is

 

[Alchymist]

Y'all are approaching it the hard way. Here's a formula you need to either memorize or make note of:

C= Dsin(A/2)
where C= chord length in a circle
D= diameter of the circle, and A= angle between points.
This is the cat's meow for laying out bolt holes on circles, etc.

Here's the approach for the above problem:

C= 20", and the angle will be 90° (4 sides in 360°).

Solve for D:

C= Dsin(A/2)

20 = D sin (90°/2)

20 = D sin 45°

20 = D (.707)

20/.707 = D

D = 28.2885"

Most all calculators have the sin function, takes about 15 seconds to solve.

Here's a little chart that's worth downloading:

 

[memphisnate]

Almost positive "A^2 + B^2 = C^2" is easier.

 

[BFBOB]

Trigonometry gives me hives.

 

[Alchymist]

Almost positive "A^2 + B^2 = C^2" is easier.

OK, lets see you use it to figure the spacing on a 9" circle for 17 bolts!
(1.654").

 

[TwoInch]

i dont get it. why wouldnt you just measure corner to opposite corner of a 20"x20" square?? would that not be the diameter of the needed circle?

am i confused? seems to easy.

 

[Quijote]

You want a circle with a diameter equal to the diagonal of a 20" x 20" square.

Diagonal of any square is (square root of 2) x (length of one side of square)

(square root of 2) = 1.414

Answer: 1.414 x 20" = 28.28"

Many ways to get there, but the cleanest and simplest is the answer above.

It's half a 20" square or an isosceles triangle with 20"-long equal sides. The base of an isosceles is 2^1/2 (square root of 2).

Square root of two is 1.4142 (known this since middle school and still remember for reasons like this).

 

[Quijote]

i dont get it. why wouldnt you just measure corner to opposite corner of a 20"x20" square?? would that not be the diameter of the needed circle?

am i confused? seems to easy.

It is easy. But if he wants an EXACT number, you are better off doing it mathematically, which is even easier.

 

[memphisnate]

OK, lets see you use it to figure the spacing on a 9" circle for 17 bolts!
(1.654").

You're correct, yours would be easier for that purpose...but not for finding the length of the hypotenuse.

 

[volvo]

..
If the city wants a pier 20x20x12. Could the answer also be, concrete total volume = to the 20x20x12?, in any shape 12" deep?

 

[Falcon67]

I'd rather know who is in the avatar

Google images "sensual wine". It's in there.

If I figure right, 4 of those is about 1500 lbs of concrete, so the post about this being used to keep the pergola on the ground in high wind is probably the deal. I would cut it like the cut the sink hole in our granite - cut most of the sides, then angle across the corners. That takes out the big chunk, then finish up with small cuts to square up the hole.