Area: Typically Defined in Square Units
Area can be defined as the amount of space a two-dimensional figure or object takes up. There are various formulas for determining area, depending on its shape. These formulas reveal the amount of area based on some unit of measurement. One of the most familiar ways to calculate area is to multiply length times width. This works for squares and rectangles. So the area of a square that is 3 inches by 4 inches is 12 square inches.
Area is typically calculated in terms of square units, such as inches, feet, meters, miles, etc. In calculating area, each side is bounded by a straight line with no width, and the area is given in terms of the number of square units the figure occupies. In physical objects such as a tract of land or the surface of a desk, the length and width are represented by geometric lines that go in a straight line along the edges.
many cases, determining area by treating the borders of a figure as having
a line with no width works well. Often
there is no issue about exactly where the border lies.
border area between one inch and two inches on a ruler is conceived as
having no width, so that the question of where to begin and stop measuring
simply does not arise. The
lines on the ruler are lined up with the object being measured, and an
inch is marked off.
Treating a boundary line as one with no width works
quite well in some cases, particularly when the
boundary line is so thin relative to what it is borders
that no purpose would be served by treating the boundary line
as having width. For example, a piece of rope that separates
two tracts of land may be so thin relative to the size of the
land that no purpose would be served by trying to specify the
boundary more precisely. Even if there is a small portion of
land that lies directly on this boundary, this portion is so
small that it can be treated as nonexistent for the purposes
of dividing the two tracts of land.
However, the case may be different when what lies on the boundary becomes important, or when the boundary line is large relative to the size of the marked area. For example, if gold lies on the boundary line between two properties, it may become important to try to specify which portion belongs to which property. Or with the center line on a highway, the line is a dividing line between the two sides of highway and doesn't belong to either side. The line is significant in size relative to the width of the road, even though it may be only several inches wide and is an example of a boundary line with important width. A doorway between two rooms provides a similar example of a boundary that has width. The area within the doorway typically doesn’t belong to either room; it is there as a three-dimensional dividing line between the two rooms.
Football provides another example where the physical boundaries have width. A football field is marked off with nine ten-yard markers. Each yard line is several inches wide. If the football rests anywhere on these lines, it is “on the ten yard line,” for example. However, at the goal line, Euclidean geometry takes over again. To score a touchdown, the player with the ball must position the ball so that it breaks the plane of the goal line before he is “down.” Here the inside edge of the goal line is treated as marking a plane with no width that the ball must cross for a goal to be scored.
In baseball, the situation is similar. Chalk lines are laid down from home plate down either line to distinguish fair territory from foul territory. These chalk lines are several inches wide. However, if a ball lands on the chalk line, it counts as a fair ball. It is only foul if it lands outside the chalk line. So in this case, the chalk line is treated as an extension of fair territory. The all important “foul pole” is really a “fair pole” since balls that hit it are considered to be in play.
Line with No Width Is No Line at All
So does it make sense to treat lines as having width? To answer this question, let us look at the function of measurement. To measure the area of an object is typically to find out how many units it contains, where the unit is some unit of length, volume, or other unit of measurement. When someone is baking a cake, for example, that person wants to know how many cups of flour to put into the cake. Likewise, quantities are important in commerce. A customer who buys a gallon of milk wants to know that she is getting one gallon, not some percentage of a gallon. One function of measurement, then, is to specify quantities for practical matters such as recipes, and to insure that people get the advertised quantities of products.
If we treat lines as having no width, this may have no practical impact in some situations. If someone wishes to divide a piece of cake into two equal slices, he or she may simply mark a line in the middle and physically divide the cake by cutting along the line. This act of division forces all particles into one side or the other, and creates two pieces of cake where formerly there was one. Of course, some "crumbs" may result which are the particles of cake that don't stick to one piece or the other; these are the byproducts of the division process.
When the quantities are not being physically divided but only divided by a line, as in the border between two towns, the width of the line may make a difference. In some cases, where the border is disputed, a no-man's-land may be specified to mark an area between two provinces or countries that belongs to neither one. For example, the Korean Demilitarized Zone (DMZ) is a 160 mile long and 2.5 mile wide border between North and South Korea. It was created as part of the armistice agreement between North and South Korea in 1953. It is a buffer zone between the two countries, It is roughly located at the 38th parallel, and is not part of either country.
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