August 24th, 2019, 01:59 PM  #1 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7  Torus Volume
Hello, So, I was helping out at a service project, and I saw a vent tube on the ceiling that was curved. it was 1/4 of a torus, to be exact. For those that don't know, a torus is like an inner tube or a doughnut shape. My goal was to find the volume of this 1/4 of a torus. After experimenting a little, I got the answer. Let's say it is a full torus. Let's then say that the longer radius, (from the middle of the torus to the middle of the tube,) is equal to 3.5 feet. The smaller radius is equal to 0.5 feet. What I did was I turned the larger radius into a cylinder, then found the volume. In this case, it would be 2.748893572........ Then, I used that volume as a radius and did 2*Pi*r to get the circumference, then divided it by 4 to get the answer I needed. In a nutshell, ((Pi*0.5^2)*3.5) * (2Pi) The volume to this 1/4 torus is 4.317951925....... I then thought to myself, "Using a cylinder volume as a radius? Ingenious, but is it correct?" I looked up the formula for the area of a torus. To my astonishment, it was similar to my formula, and it got the same answer. This formula was (Pi*r^2)*(2*Pi*R) where r equals the smaller radius, and R equals the larger radius. I input my data. (Pi*0.5^2)*(2*Pi*3.5) 0.785398163....*21.99114858.... 17.2718077...... Then I divided it by four, because it was 1/4 of a torus I needed. 4.317951925....... I was right. I thought to myself the following: "Jared! One of your crazy math ideas WORKED!" Well, it was fun learning about torus volume. I enjoyed it. I am beginning to get good at geometry! Math is so fun! I wanted to share this with you all. I hope you enjoy it. Jared Last edited by skipjack; August 25th, 2019 at 04:46 AM. 
August 24th, 2019, 03:25 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600 
Second theorem of Pappus ... 
August 24th, 2019, 03:47 PM  #3 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
@skeeter, thanks for sharing that with me. Is that what this is called? The second theorem of Pappus?

August 25th, 2019, 04:31 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600 
Link to the first and second theorems of Pappus ... Pappus's Centroid Theorem  from Wolfram MathWorld 
August 25th, 2019, 04:58 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222  
August 25th, 2019, 07:46 AM  #6 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
@skipjack, sure. Here is what I did. The length from the middle of the torus to the middle of the tube is 3.5. The radius of the tube is 0.5. I did the following with these numbers to turn the larger radius into a cylinder with a 0.5 radius and a 3.5 length. Pi*0.5^2 After that, I multiplied it by the larger radius, 3.5. The radius of the torus is now the volume of a cylinder assuming the radius is a 3d cylinder. Simply multiply that by 2pi and you get the torus volume. Divide it by 4 to get the area I was looking for. Does this explain it well enough? Jared 
August 25th, 2019, 08:47 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222 
Your first step produced $\pi$0.5². This was presumably based on the formula for the area of a circle, but you didn't mention this circle or why you chose to use it. Your second step was to multiply by the larger radius, 3.5. That doesn't mean that that the multiplier "is now a volume" or "has turned into a cylinder". Rather, you have used it as the length of a cylinder, but why did you choose that particular cylinder? You seem also to have assumed, without saying so, the formula for the volume of a right circular cylinder. 
August 25th, 2019, 10:12 AM  #8 
Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 
The dude correctly deduced the formula for the volume of a torus through visualisation. What are you harping on him for?

August 25th, 2019, 11:55 AM  #9 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
one thing you need to know about me, I think a lot differently than most people. It comes with ADHD. I visualized the whole thing in my head, then I figured it out. Basically, I think in pictures. That plus my calculator. Regardless, I got the same answer as the correct formula. Anyway, just for the record, I don't feel he's getting on my case. I don't feel offense at all. I have an idea that might help explain my method. Just give me some time. Jared 
August 25th, 2019, 12:27 PM  #10 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7  I made a video of how I did it with me explaining what I did. I hope this helps! Jared 

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torus, volume 
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