Perimeter of a triangle with rounded corners

[thedge777]

Hello, I'm trying to build an equation that I can use to calculate the perimeter of a triangle with rounded corners. I'm not looking for a specific answer but an equation so I can plug different numbers in to calculate the perimeter for different sizes.

The actual issue is I'm trying to bundle 3 cables together, two larger ones horizontally and one smaller one on top. I want to find the exact length of a piece of rope or something wrapped around the entire bundle. The attached image shows this. I'm trying to calculate the perimeter of the solid line and the dashed lines are just to indicate where the cables would be. For the attached example, the cables on the bottom have a 3" radius and the top cable has a 1.5" radius.

I worked out how to find the length of the tangent lines between them using this equation and it seems to work well:

[math]L = \sqrt{{(D - (R1 + R2))^2 - (R2 - R1)^2}}[/math]
L is length of tangent line
D is the distance between the center points of the circles
R1 and R2 are circle radii

The part I am stumped on is calculating the length of the arcs between the tangent lines. I can do it if all three cables are the same size by just taking 1/3rd of the circumference but that all changes when I have multiple cable sizes.

Hope this is clear enough. Any help will be greatly appreciated. Thank you!

 

[Dr.Peterson]

I worked out how to find the length of the tangent lines between them using this equation and it seems to work well:

[math]L = \sqrt{{(D - (R1 + R2))^2 - (R2 - R1)^2}}[/math]
L is length of tangent line
D is the distance between the center points of the circles
R1 and R2 are circle radii

The part I am stumped on is calculating the length of the arcs between the tangent lines. I can do it if all three cables are the same size by just taking 1/3rd of the circumference but that all changes when I have multiple cable sizes.

Before I work on the arcs, can you show how you derived your formula for L? I'm not sure I understand it, or that it is correct.

Is D = R1 + R2, or something else? Is L one of the sloped segments in the picture?

I get a much simpler formula for that:

 

[thedge777]

I'm sorry... I wrote that out wrong.

This is the one I used:
[math]L = \sqrt{{(D^2) - ((R1 - R2)^2)}}[/math]
You are right that D=R1+R2 which means in the first formula I posted, I literally had D-D as part of my formula...

Also, yes, L is one of the sloped tangent lines.

I don't fully recall how I came up with it but I was trying to do it thinking in terms of triangles and was trying to use cosine. I went through so many different variations of it before I was able to get the correct answer that I'm not sure all of exactly what I did. The only reason I knew I had the correct answer is I have it drawn up in CAD where I can measure it and compare that to what my formula is putting out.

When I put your much simpler formula into my Excel sheet, it is coming up with the exact same answer as mine, so my formula is giving the correct answer but is just way over-complicated.

 

[The Highlander]

The actual issue is I'm trying to bundle 3 cables together, two larger ones horizontally and one smaller one on top. I want to find the exact length of a piece of rope or something wrapped around the entire bundle. The attached image shows this. I'm trying to calculate the perimeter of the solid line and the dashed lines are just to indicate where the cables would be. For the attached example, the cables on the bottom have a 3" radius and the top cable has a 1.5" radius.

Wouldn't it be easier just to use a measuring tape? ?

              

 

[thedge777]

Wouldn't it be easier just to use a measuring tape? ?

               View attachment 36069

I have it drawn in CAD so I already know the length of this example. However, I need a formula that I can plug in different cable sizes and get the amount of rope to wrap in a spiral pattern around the bundle. I have all the math figured out except finding the perimeter…

 

[blamocur]

Before I work on the arcs, can you show how you derived your formula for L? I'm not sure I understand it, or that it is correct.

Is D = R1 + R2, or something else? Is L one of the sloped segments in the picture?

I get a much simpler formula for that:

View attachment 36064

An irrelevant comment: I've never seen before a geometric construction of geometric mean.

 

[Dr.Peterson]

I have it drawn in CAD so I already know the length of this example. However, I need a formula that I can plug in different cable sizes and get the amount of rope to wrap in a spiral pattern around the bundle. I have all the math figured out except finding the perimeter…

Yes, that's my understanding of what you want.

I'm sorry... I wrote that out wrong.

This is the one I used:
[math]L = \sqrt{{(D^2) - ((R1 - R2)^2)}}[/math]
You are right that D=R1+R2 which means in the first formula I posted, I literally had D-D as part of my formula...

Also, yes, L is one of the sloped tangent lines.

I don't fully recall how I came up with it but I was trying to do it thinking in terms of triangles and was trying to use cosine. I went through so many different variations of it before I was able to get the correct answer that I'm not sure all of exactly what I did. The only reason I knew I had the correct answer is I have it drawn up in CAD where I can measure it and compare that to what my formula is putting out.

When I put your much simpler formula into my Excel sheet, it is coming up with the exact same answer as mine, so my formula is giving the correct answer but is just way over-complicated.

Your corrected formula simplifies to mine, so you probably got it more or less the way I did, using triangle BCM below.

[math]L=\sqrt{{D^2 - (R_1 - R_2)^2}} = \sqrt{{(R_1 + R_2)^2 - (R_1 - R_2)^2}}=\sqrt{(R_1^2+2R_1R_2 + R_2^2) - (R_1^2-2R_1R_2+R_2^2)}=\sqrt{4R_1R_2}=2\sqrt{R_1R_2}[/math]
As for the arcs, before I show you a (rather ugly) formula, or set of formulas, I'd like to see if I can get you to work them out yourself. The basic method is to use triangle BCD to find angle [imath]\alpha[/imath], and to use triangle BCM to find angle [imath]\beta[/imath]:

​

Then the angles for the arcs can be calculated from that. I tried making a single formula, and it's too complicated to bother with.

An irrelevant comment: I've never seen before a geometric construction of geometric mean.

Here's a standard one, plus others (including the one we're discussing):

Geometric Meaning of the Geometric Mean

www.cut-the-knot.org

 

[blamocur]

Then the angles for the arcs can be calculated from that. I tried making a single formula, and it's too complicated to bother with.

I don't know if it is important in this particular application to consider the case when [imath]r[/imath] is too small to touch the perimeter.

 

[blamocur]

Yes, that's my understanding of what you want.


Your corrected formula simplifies to mine, so you probably got it more or less the way I did, using triangle BCM below.

[math]L=\sqrt{{D^2 - (R_1 - R_2)^2}} = \sqrt{{(R_1 + R_2)^2 - (R_1 - R_2)^2}}=\sqrt{(R_1^2+2R_1R_2 + R_2^2) - (R_1^2-2R_1R_2+R_2^2)}=\sqrt{4R_1R_2}=2\sqrt{R_1R_2}[/math]
As for the arcs, before I show you a (rather ugly) formula, or set of formulas, I'd like to see if I can get you to work them out yourself. The basic method is to use triangle BCD to find angle [imath]\alpha[/imath], and to use triangle BCM to find angle [imath]\beta[/imath]:

View attachment 36073​

Then the angles for the arcs can be calculated from that. I tried making a single formula, and it's too complicated to bother with.


Here's a standard one, plus others (including the one we're discussing):

Geometric Meaning of the Geometric Mean

www.cut-the-knot.org

Is the sign of [imath]a_1[/imath] wrong? I.e., should it be [imath]a_1 = 2(\alpha+\beta) - 180[/imath] ?

 

[Dr.Peterson]

Is the sign of [imath]a_1[/imath] wrong? I.e., should it be [imath]a_1 = 2(\alpha+\beta) - 180[/imath] ?

Yes, I did that part in my head and forgot to check it.

 

[thedge777]

I

I don't know if it is important in this particular application to consider the case when [imath]r[/imath] is too small to touch the perimeter.

It’s not important because, in that case, I would just calculate the perimeter of the two larger cables and ignore the smaller one.

 

[thedge777]

Ok... So this is how I came to the final answer:

First, I calculated the length of the tangent lines using the formula given me:
[math]2 \cdot \sqrt{R \cdot r}[/math]
Then, following the hints from Dr. Peterson, I solved angles α and β and subtracted them from 270 to get angle B (as pointed out on the drawing). This gave me the information I needed to calculate the circumference of the larger arcs with this formula:
[math]\left(\frac{B}{360}\right) \cdot 2\pi R[/math]
I then used the same process to find the circumference for the arc on the smaller circle. I just subtracted angles A and B from 360 to get angle C which I used the above formula with to get the answer.

After this, I just added up all of the arcs and tangent lines to get my final answer.

Thank you so much, Dr. Peterson, for your help! Let me know if I got any of the above wrong.

 

[Dr.Peterson]

I just subtracted angles A and B from 360 to get angle C

Yes, that's equivalent to what we had (eventually) said,

Is the sign of [imath]a_1[/imath] wrong? I.e., should it be [imath]a_1 = 2(\alpha+\beta) - 180[/imath] ?

That is, [math]C=360-2A=360-2(270-\alpha-\beta)=360-540+2(\alpha+\beta)=2(\alpha+\beta)-180.[/math] And it's a nice, efficient way to get that result, since you had to find A and B already. I like it.