Pappus' Theorem (surface area)

In summary, the conversation discusses using Pappus' Theorem and the surface area of a sphere to find the centroid of a semicircle defined by the equation x= sqrt ( c^2 - y^2). The conversation mentions the equation S = 2 (pi) * p * L, where S is surface area, p is the distance from the axis of revolution, and L is the length of the arc. The individual seeking help is unsure of how to proceed and is encouraged to think about the meaning of the variables in the equation. A link to a reference on Pappus' Theorem is also provided.
  • #1
whatlifeforme
219
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Homework Statement


Use Pappus' Theorem for surface area and the fact that the surface area of a sphere of radius c is 4(pi)c^2 to find the centroid of the semicircle x= sqrt ( c^2 - y^2)


Homework Equations


S = 2 (pi) * p * L

where s=surface area; p=distance from axis of revolution; L= length of the arc


The Attempt at a Solution


1. centroid of semicircle. should i put the equation in circle form, and attempt to solve from that?
 
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  • #2
whatlifeforme said:

Homework Statement


Use Pappus' Theorem for surface area and the fact that the surface area of a sphere of radius c is 4(pi)c^2 to find the centroid of the semicircle x= sqrt ( c^2 - y^2)

Homework Equations


S = 2 (pi) * p * L

where s=surface area; p=distance from axis of revolution; L= length of the arc

The Attempt at a Solution


1. centroid of semicircle. should i put the equation in circle form, and attempt to solve from that?

No, you don't. You need to think about what that equation means and how it's related to what you want to find. It's almost all you need to know.
 
  • #3
i'm still lost.
 
  • #4
whatlifeforme said:
i'm still lost.

Explain to me what the variables in that equation mean. Yes, S=surface area. But surface area of WHAT? Look it up if you don't have a good reference handy.
 
  • #5
S = 2 (pi) * p * L

where s=surface area; p=distance from axis of revolution; L= length of the arc
 
  • #6

FAQ: Pappus' Theorem (surface area)

1. What is Pappus' Theorem (surface area)?

Pappus' Theorem (surface area) is a mathematical theorem named after the Greek mathematician Pappus of Alexandria. It states that the surface area of a three-dimensional object can be found by multiplying the length of its circumference by the distance traveled by its centroid while rotating around an axis.

2. How is Pappus' Theorem (surface area) used?

Pappus' Theorem (surface area) is commonly used in calculus and geometry to find the surface area of objects with curved surfaces, such as spheres, cones, and tori. It can also be used to calculate the surface area of more complex shapes by breaking them down into simpler components.

3. What are the requirements for using Pappus' Theorem (surface area)?

In order to use Pappus' Theorem (surface area), the object must have a constant cross-sectional area and its centroid must travel a constant distance while rotating around an axis. Additionally, the axis of rotation must be parallel to the cross-sectional area.

4. Are there any limitations to Pappus' Theorem (surface area)?

Yes, Pappus' Theorem (surface area) is limited to finding the surface area of objects with continuous and symmetrical cross-sections. It also cannot be used to find the surface area of objects with changing cross-sectional areas or non-uniform rotations.

5. Can Pappus' Theorem (surface area) be applied to any shape?

No, Pappus' Theorem (surface area) can only be applied to objects with rotational symmetry. This means that the shape of the object must remain the same after being rotated around an axis. Examples of shapes that can be used with Pappus' Theorem include cylinders, spheres, and cones.

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