Confusion regarding area of this figure

  • Context: High School 
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    Area Confusion Figure
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Discussion Overview

The discussion revolves around the calculation of the area of a closed figure composed of two parallel lines and two semicircles, one inward and one outward. Participants explore different reasoning approaches to determine the area, considering both geometric interpretations and potential misconceptions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the area of the figure can be calculated as A=2rb, arguing that the area lost by the inward semicircle is compensated by the outward semicircle.
  • Another participant challenges this reasoning by stating that the figure is flat and that adding semicircles does not apply in this two-dimensional context.
  • A different viewpoint is presented, questioning the validity of using infinite semicircles to derive an area of pi*r*b, suggesting that this reasoning leads to contradictions.
  • One participant emphasizes the dimensionality of the space, arguing that the reasoning used to calculate area in a curved space does not apply to the flat figure in question.
  • Another participant raises a question about the validity of adding curve lengths in two dimensions, seeking clarification on why this approach is deemed incorrect.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correct method to calculate the area of the figure. Multiple competing views remain, with no consensus reached on the validity of the various reasoning approaches presented.

Contextual Notes

Participants note that the reasoning involves assumptions about dimensionality and the nature of curves in two-dimensional space, but these assumptions are not resolved within the discussion.

Prem1998
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figure.png
Maybe, it's a useless question. The figure which I'm talking about consists of two parallel lines each of length 'b' and are separated by a distance 2r. Their ends on one side is closed by a semicircle which in pointing inwards and decreases the area and the ends on the other side are joined by another semicircle of the same radius 'r' but this time it is bulging outwards and contributes to the area. So, we have a closed figure. Finding its area is simple because the amount of area decreased by the first semicircle gets added again by the second semi-circle. So, the area remains the same as that of a rectangle with length 2r and breadth b. A=2rb.
But I get a different answer by this reasoning:
The given figure can be thought to be made up of an infinite number of semicircular arcs from top to bottom. The figure is filled with semicircles. So, the area of this figure can be thought to be the sum of the lengths of these infinite number of semicircles. The length of each elementary semicircle, i.e. pi*r is constant. And, these semicircles are distributed over a length 'b'. So, the area of the figure = pi*r*b, which is wrong. But, what is wrong with this reasoning?
 
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The fact it is not three dimensional. Your adding semicircles, but actually you only have flat straights. And you stretched these straights to a semicircle. If this were correct, you could create even larger numbers by choosing longer curves than a semicircle.

In space as a surface of a cylinder you were right, because there is no shorter way than going around. But this one is flat.
 
fresh_42 said:
If this were correct, you could create even larger numbers by choosing longer curves than a semicircle.
I know it's not correct and that this reasoning would lead to contradictions.. But, what is wrong with the points I've made to prove that the area should be pi*r*b. I mean, consider a rectangle of length 'pi*r' and breadth 'b'. This rectangle also has an infinite number of 'pi*r lengths' distributed over a breadth 'b'. But, this rectangle has an area pr*r*b. Even if the number of semicircular arcs in the figure is infinite, but a breadth 'b' should be able to accommodate the same number of lines both in the this rectangle of length 'pi*r' and the figure that I'm talking about. After all, both the semicircular arcs in the figure and the straight lines in this rectangle have breadth equal to that of a point, i.e zero (according to definition).
 
Prem1998 said:
But, what is wrong with the points I've made to prove that the area should be pi*r*b.
fresh_42 said:
The fact it is not three dimensional.

You are calculating in a space with curvature.

Untitled.png


You extended the green line which you should calculate with, to become the read line. Why the red one? Why not the blue one?
 
fresh_42 said:
The fact it is not three dimensional.
How does this lead to the conclusion that I can't add semicircles? Where is it proved that I can't add curve lengths in two dimensions?
 

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