My Math Forum Torus Volume

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 August 25th, 2019, 03:11 PM #11 Global Moderator   Joined: Dec 2006 Posts: 21,036 Thanks: 2274 You were clearly aware of, and used, the standard formulas for the area of a circle in terms of its radius, and the circumference of a circle in terms of its radius. You said "I thought to myself for the volume of a torus... it's basically a circular cylinder." I would take that to mean that you thought of the torus as an "ordinary" (straight) cylinder bent into the shape of a torus. The length of that straight cylinder would be the length of the circumference of a circle of radius R, which is 2$\pi$R, so the volume of the torus would then be $\pi$r² (the area of each end of the cylinder) multiplied by 2$\pi$R (where r and R are the two radii that you defined in the video), i.e. ($\pi$r²)(2$\pi$R). That's the correct formula, obtained in what you later called the normal way, but you said that you obtained it slightly differently - in two steps. First, you obtained ($\pi$r²)(R), the volume of a cylinder of length R, using the known values of r and R, and then you multiplied that by 2$\pi$, explaining that you were turning R into the (corresponding) circumference. Your two steps reached the same formula, but the only explanation you gave for using your two-step process was "because that's the way I visualized it". That's the specific thing that I didn't understand. Having visualized the torus as "basically a circular cylinder", why did you decide on a two-step process for finding its volume?
 August 25th, 2019, 05:03 PM #12 Member   Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 Well, here's why I decided it. At first when I was working on this, I thought to myself, "it is basically a circular cylinder" as you said. I did the two step process of finding the cylinder volume radius first and then finding the circumference because I felt like it would be accurate enough to calculate the volume of the Torus. I felt it would have worked, which it did. Yes, it is a two-step process, but it's just as accurate as what I call "the normal way." I guess some people have different ways of figuring out the same math problem. Jared
 August 26th, 2019, 03:08 AM #13 Global Moderator   Joined: Dec 2006 Posts: 21,036 Thanks: 2274 You used your judgement and succeeded. Let's see what you make of the following. Confusion.PNG My sketch above relates to a rectangle ABCD, with lines DE and BE added so that BE is equal in length to BC and AD. The point E is chosen so that angle CDE is fairly small, but not zero, and DE is longer than DC. Hence points C and E do not coincide. The points F and G are the midpoints of DE and AB respectively. As DE and AB are non-parallel, the perpendiculars to them through their midpoints are non-parallel and must meet, as at P (see FP and GP in the diagram). I've added the dashed lines PA, PB, PD and PE. AD and BE are equal in length (as already stated). Triangle DEP is symmetrical about PF, so PD and PE are equal in length. Triangle ABP is symmetrical about PG, so PA and PB are equal in length, and the angles PAG and PBG are equal in magnitude. Hence triangles ADP and BEP are identical in shape and size (congruent), because all their corresponding sides match each other in length. It follows that angles DAP and EBP are equal in magnitude. From those equal angles, subtract the equal angles PAG and PBG respectively, leaving angles DAG and EBG (see diagram). However, angle DAG is a right angle (because ABCD is a rectangle), whereas angle EBG is not a right angle (else points C and E would coincide). Which, if any, of my steps above doesn't seem correct to you?
 August 26th, 2019, 05:36 AM #14 Member   Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 I will work on that problem later today, if that is ok. I have to fix my car so I can go to the dentist this morning. I'll give an answer when I can. Jared Last edited by skipjack; August 26th, 2019 at 09:24 AM.
 August 26th, 2019, 08:47 AM #15 Member   Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 Okay, I just worked on it. I apologize for the delayed response. After reading through it, and drawing on a whiteboard, right now I conclude that everything you said, and the process, made sense to me. To be honest, I didn't find anything wrong with it. However, to be honest, I don't know if I'm correct or not. I'm going to be thinking about it a little bit more. I did understand everything you said though. Came through very clear to me. I'm going to keep working on this.
 August 26th, 2019, 01:06 PM #16 Member   Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 @skipjack, I have to say one thing. You stumped me this time. I'm not finding anything wrong with everything you gave me in that math problem. I got it on my whiteboard, looking at it, and I see nothing wrong. That is the best I can conclude, that there is nothing wrong. Am I correct? If not, it looks like I still have some more stuff to learn! Jared

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