# How Does the Pappus-Guldinus Theorem Apply to Calculating Volume?

• suspenc3
In summary, the conversation discusses the calculation of volume and mass using the Pappus-Guldinus Theorem. The discussion includes different approaches and suggestions for finding the volume and applying the theorem, as well as a brief mention of the term "Megagram."
suspenc3
http://img244.imageshack.us/img244/9114/centroidsdq6.th.jpg

The book only briefly covers this section and there is an example with a sphere, but I don't really know how to get started.

Heres what I tried:

I basically cut the upper region up into three rectangles, two of them having negative area:

triangle 1)$$A=240(100)=24000mm^2$$
$$\bar{y}=250mm$$
Distance traveled by C $$2\pi(250)=1571mm$$
Volume = $$1571mm(24000mm^2=37704000mm^3$$

And then the same thing was done for the other two rectangles of negative area, and then a total volume was found. From this I found a mass which I think is wrong.

Is this the right approach?

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I don't really know what the Pappus-Guldinus Theorem is or how to apply it, but couldn't you find the volume by using the differences in areas of circles for the each of the sections of different thickness multiplied by the thickness? Just a thought, maybe it might be useful as a check to see if you get a consistent answer.

Yeah I could try that, I'd still like to know how to apply this theorem though

Sorry, I'm not familiar enough with it to be of much use. I'd be more helpful if I could.

I looked it up; if I understand it is the Vol=planform times the distance the centroid would travel in sweeping out the object.

What I tried is to break into the three rectangles the centroids of each should be easy to figure by symmetry.

I get 2pi(30*100*(80+15)+250*30(110+125)+60*100*(360+30))=
2pi(3000*95+7500*235+6000*390)=
2pi(285,000+1,762,500+2,340,000)=27,567,475.5mm^3=
.0275m^3

ahhhhh, yeah that makes sense!

how many kilos are in a Mg?(Megagram?)

1000?

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i have never heard of such a creature, my guess would be the same.

## What is the Pappus-Guldinus Theorem?

The Pappus-Guldinus Theorem is a geometric principle that relates the volume or surface area of a solid or surface of revolution to the distance traveled by its centroid. It is named after ancient Greek mathematician Pappus of Alexandria and the 16th century mathematician Paul Guldinus.

## What is the formula for Pappus-Guldinus Theorem?

The formula for the Pappus-Guldinus Theorem is V = 2πA̅d, where V is the volume of the solid or surface of revolution, A̅ is the area of the generating region, and d is the distance traveled by the centroid of the generating region.

## What are some real-world applications of Pappus-Guldinus Theorem?

Pappus-Guldinus Theorem has many applications in engineering and physics, such as calculating the volume of a wine barrel, the surface area of a propeller blade, or the volume of a water tank with a curved base. It is also used in calculating the moment of inertia and center of mass of objects with rotational symmetry.

## How is Pappus-Guldinus Theorem related to other mathematical concepts?

Pappus-Guldinus Theorem is closely related to other mathematical concepts such as calculus, geometry, and physics. It is often used in conjunction with the method of shells in calculus to find the volume of irregularly shaped objects. It also has connections to the parallel axis theorem and the theorem of parallel axes in physics.

## Are there any limitations to Pappus-Guldinus Theorem?

While Pappus-Guldinus Theorem is a powerful and widely applicable principle, it does have some limitations. It can only be applied to objects with rotational symmetry, and the centroid of the generating region must lie on the axis of rotation. It also assumes that the generating region is a continuous curve, which may not always be the case in real-world applications.

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