Geometry and the Globe
You draw two circle one in a plain paper and the other on a north pole, with the pole as a center. Which circle will have a bigger area and Why ?
Labels: outside-the-box
posted by Rajesh Lal at
9:11 AM
14 Comments:
It'll be the same. Else the circle on the north pole won't be a circle. More of amoeba.
The one with the larger radius?
what ever one i draw a larger circle will have the larger area.. doesn't say they have to be the same size :)
the 1 i draw bigger LOL :)
because the earth is a sphere (more or less) the circle that u did not draw on the north pole will be the bigger one, save for the one on the south pole, which would be the same size
speed
The one on the north pole will be there larger(in terms of area on the ground)
Imagine yourself walking from the center of either circle.
If you are walking on just a sheet of paper, the distance you have to walk is r- the radius.
If you are walking with the center as the north pole, your radius that you have to walk is going to be slightly bigger because its not just going as a straight horizontal, it is also moving downward.
Therefore, if the area of a circle is pi*r^2, the circle with the bigger radius (the one on the north pole) will be the bigger one.
I suppose another way to think of it would be to imagine each circle as a sheet, like a bedsheet. If one of the bedsheets is stretched tight (the circle on the paper) then obviously the sheet that is the same size around but is still loose enough to drape down like a semi circle is the bigger one. Which is essentially what the circle drawn on the north pole is doing.
Think of it another way.
If you draw a circle at the North Pole, centered on the pole you are actually drawing on a sphere; the Earth. When you draw a circle on a sphere you are actually drawing a line that splits the sphere into two circular (spherical) surfaces. One circle is centered on the North pole and the other centred on the South Pole.
This is easier to envisage if you imagine that the first circle is drawn far enough away from the North Pole so that it is actually drawn along the equator. In that case it is easy to see that this line is also centered on the South Pole, and both circles enclose the same area.
Now if we move that line northward the area enclosed by the North Pole based circle gets smaller, but the South Pole circle gets larger.
So: if you draw a small circle centered on the North Pole you could logically argue that the area that you want to consider is actually the one that contains the South Pole and is therefore MUCH larger than the one drawn on paper.
I'm actually very sorry for including so many "actually"s in my last post
A circle is a plane object, no matter if you draw in on a sphere or a plane. Drawing it on a sphere does not make the area bigger, because the area is a plane and cuts thru the surface of the sphere. If you talk about the "surface area" enclosed within the circumference, that's different.
A circle, with the earth's radius would include the 3 dimensional surface area of the whole northern hemisphere, but the area of the plane circle would be a plane cutting the center of the earth.
The problem does not explicitly state it, but the implication is that the circle centered on the North Pole is the 'same size' as the one on paper.
I can think of three different (reasonable, in my opinion) ways of defining 'same size' and two different ways of defining 'area' in the context of this problem.
Area can be defined as:
1. area of the subset of the plane, which contains the circle, that is bounded by the circle. The plane, of course, cuts through the sphere.
2. area on the surface of the sphere North of the circle.
'same' size circle can be constructed in the following ways:
1. by taking a closed loop of (infinitisemally thin) string the same length as the circle on paper and stretching it over the sphere, centered at the North Pole. This will have the same circumference as the paper circle and the same radius on the plane, that cuts the sphere, as in (1) in area variants. It will have the same area(1) and larger area(2).
2. by taking a string of the same length, as the radius of the paper circle, fixing one end of it at the North Pole and tracing the opposite end at maximum stretch all around the sphere. This will have the same radius on the sphere surface as the paper circle. It's area(2) will be the same as that of the paper circle and area(1) will be smaller.
3. by taking a compass set to the size of the paper circle's radius and using it to draw the circle centered at the North Pole. This will result in a circle, all of the points of which are at the radius distance from the North Pole as measured in a straight line in three-space (i.e. cutting through the sphere). In this case area(1) will be smaller, than that of the paper circle, and area(2) will be larger.
All of the above assumes that the circle is no larger in diameter, than the sphere, otherwise you have a different set of degenerate cases (with various degrees of degeneracy), which I will dismiss as not interesting...
I think, the above six answers exhaust all the possibilities. And me along with them :)
The circle drawn on the piece of paper will be larger because the one drawn on the north pole will blow away in the wind or will be filled with snow.
if the circle on the paper and the circle on the north pole are of same diameter...then the circle on the paper would have the larger area...it doesnt have a pole taking up some of the space inside!!!!
if the circle on the paper and the circle on the north pole are of same diameter...then the circle on the paper would have the larger area...it doesnt have a pole taking up some of the space inside!!!!
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