My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Thanks Tree1Thanks
Reply
 
LinkBack Thread Tools Display Modes
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.
jnicholes is offline  
 
August 24th, 2019, 03:25 PM   #2
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,016
Thanks: 1600

Second theorem of Pappus ...

skeeter is offline  
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?
jnicholes is offline  
August 25th, 2019, 04:31 AM   #4
Math Team
 
skeeter's Avatar
 
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
skeeter is offline  
August 25th, 2019, 04:58 AM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 20,972
Thanks: 2222

Quote:
Originally Posted by jnicholes View Post
I turned the larger radius into a cylinder, then found the volume.
Can you give the details of that?
skipjack is offline  
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
jnicholes is offline  
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.
skipjack is offline  
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?
Thanks from Greens
DarnItJimImAnEngineer is offline  
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
jnicholes is offline  
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
jnicholes is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
torus, volume



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
From band to torus Loren Topology 1 March 4th, 2017 03:52 PM
(Torus/Volume) saman65 Calculus 4 November 5th, 2015 07:51 AM
torus surface tejolson Topology 1 February 21st, 2015 07:42 AM
Torus T^2 homeomorphic to S^1 x S^1 Math Amateur Real Analysis 0 March 19th, 2014 05:25 PM





Copyright © 2019 My Math Forum. All rights reserved.