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I am thinking of a way to find the surface area of a sphere, and what i thought is , if you take a sphere whose radius = $r$ , you can think that the sphere consists on infinite circles of radiuses $r$. Just imagine that you’re rotating a circle of radius $r$ and getting a sphere of the same radius.

So you get , $2\pi r + 2\pi r + 2\pi r ......= 4\pi r^2$ .

Is this approach correct?

Widawensen
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Aaryan Dewan
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4 Answers4

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The question hinges on the fundamental questions about how area is measured mathematically. Quoting from a comment by the question asker,

If you add up the points, it must make the surface area.

That's a quite natural starting point for an attempt to define the measurement of area. But it does not work, because every sphere has the exact same number of points. To get a sphere twice as large, just move every point twice as far from the center.

In fact, there are the same number of points on the sphere as there are in an entire infinite plane or in all of three-dimensional Euclidean space, for much the same reasons as that a line segment and an entire infinite line have the same number of points or that an infinite line and an infinite plane have the same number of points or even that an infinite line and the entire three-dimensional space have the same number of points. (I'm considering only Euclidean geometry here because it's the geometry that the question seems to want to use.)

So how do we measure area? In particular, knowing that every common object whose area we could measure has the same number of points, how can we define area at all? That's an important question with a standard mathematical answer. Basically, in addition to a collection of points, you need certain concepts of distance and shape. This is how we can get a shape with length, area, or other dimensions from a collection of points that have none of those properties.

So there are some reasons (including links to other questions and answers in MSE) why you would not add lengths to get areas, as well as links to some questions and answers that discuss how you actually can measure area.

Community
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David K
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Not at all. Repeatedly summed lengths can never make up an area. To get an area you need to multiply a length with a length, but even side-wise addition would not do.

Narasimham
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  • Why? Please Give an appropriate reason before giving answers. Area means, the surface it covers, in other words the points. It you add up the points. it must make the surface area. – Aaryan Dewan Mar 05 '16 at 17:24
  • @AaryanDewan The problem is that the area of one point is zero. You cannot literally add up infinitely many zeros to get a finite nonzero number. Instead you need some kind of approximation procedure, where you approximate the area by finitely many rectangles, sum their areas, and then increase the number of rectangles. This is what integration (a topic from calculus) achieves. – Ian Mar 05 '16 at 17:47
  • @Aaryan Dewan as noted above. – Narasimham Mar 05 '16 at 18:04
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That way is not correct. Rather put the sphere in a 3-D coordinate system and make an arbitrary circle on it whose area vector is pointing in z direction. Now move a little bit on the sphere and you have a circular strip. Its area will be $2 \pi xdc$ $dc=Rd\theta$ and $ x=Rcos\theta$ Integrate from $0$ to $\pi$ and you get the surface area of the sphere.

user41736
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Your propoed answer is incorrect, you can do it as follows instead; $$S=\int_{0}^{\pi} \int_{0}^{2\pi} Rd\theta R sin(\theta) d\phi \\ \ \ =-R^2 [cos(\pi)-cos(0)](2\pi) \\ \ \ =4 \pi R^2$$

user281270
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