Area of a Sphere: Intuitive Problem Solved

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Discussion Overview

The discussion revolves around the intuitive understanding of the area of a sphere, particularly how it relates to the geometry of a circular ring and the concept of rolling to form a sphere. Participants explore the mathematical derivation of the sphere's area and challenge each other's reasoning regarding the area calculations.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about deriving the area of a sphere from the circumference of a circular ring, suggesting that rolling the ring results in an area of \(2 \pi^2 R^2\).
  • Another participant questions the validity of the multiplication by \(\pi R\) in the initial reasoning, implying that there is a standard method for calculating areas that should be followed.
  • A participant introduces the example of a cylinder to illustrate how area is calculated by rotating a line segment, emphasizing that this method cannot be directly applied to derive the area of a sphere.
  • Further, a participant discusses the area of strips cut from the northern hemisphere of the Earth, noting that these strips do not form cylinders and that their areas vary due to the cosine factor at different latitudes.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial reasoning regarding the area of a sphere. There are competing views on how to properly derive and understand the area, with some participants challenging the methods proposed by others.

Contextual Notes

Participants highlight limitations in the reasoning presented, particularly regarding the assumptions made about the shapes involved and the implications of rolling a circular ring to form a sphere. The discussion reveals a dependence on geometric definitions and the need for careful consideration of the shapes being analyzed.

ChrisVer
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I am sure this should have already be discussed somewhere in the past...
I have an intuitive problem with the area of a sphere. Following the mathematics of the metric and surfaces, I can easily derive the area of a sphere which is 4 \pi R^{2}.
Now I'm have this problem:
Suppose I get a ring [circular] of radius R... then it's circumference is 2πR okay?
Then how could someone create a sphere? just roll it by an angle π around itself [check attachement]... But if I say so, won't the area be (2 \pi R) \times (\pi R) = 2 \pi^{2} R^{2} ?

then one could say that each point on the circle is not going to cover a πR rotation, but a πr (r<R) ... double time because for each point which covers this rotation,there exists a symmetric one which covers the same... I think even with that, the area covered will be larger than the one given for a sphere.
 

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I don't know how you got that multiplying by \pi R stuff but here's how its usually done.
 
Shyan said:
I don't know how you got that multiplying by \pi R stuff but here's how its usually done.

what I mean is take for example a cylinder of radius R and length L...
Then what's its area? it's L *2πR , because you take the line of length L and rotate it by 2pi around a center
 
ChrisVer said:
what I mean is take for example a cylinder of radius R and length L...
Then what's its area? it's L *2πR , because you take the line of length L and rotate it by 2pi around a center

The point you're missing is that you can't deform that cylinder to a sphere!
 
ChrisVer said:
what I mean is take for example a cylinder of radius R and length L...
Then what's its area? it's L *2πR , because you take the line of length L and rotate it by 2pi around a center

Suppose you cut the northern hemisphere of the Earth into strips:

  1. from 0° latitude to 1° latitude
  2. from 1° to 2°
  3. etc.

What is the (approximate) area of each strip? First thing you should notice is that these strips are NOT cylinders, because one edge is slightly smaller than the other. The distance around the Earth at latitude \theta is 2\pi R cos(\theta). So it's only 2\pi R at latitude 0°.

Since the strips aren't the same size, you can't just multiply by the number of strips.
 

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