INSTRUCTIONAL UNIT: Distance, Area, and Volume
Distance, area and volume are progressively higher dimensional versions of the same thing: measurement. Distance is a measurement of one dimension, area is a measure of two dimensions and volume is a measure of three-dimensions. We will later explore measurements in four-dimensions. However, let us first turn our attention to the most basic measure, distance.
Distance is the unit measure between two points. The shortest distance between two points in the Euclidian plane is a straight line, although this shortest distance may be an arc-length, on the surface of a sphere For example, the shortest distance between New York and Los Angeles is approximately 2,781.81 miles, but this distance would be shorter if we tunneled through the earth. This distance represents the length of an arc on the surface of the earth.
The shortest distance between two points in the Euclidian plane however, is a straight-line segment (Figure 1). To find the distance between points A and B is to measure the length of the line segment.
Figure 1: The distance between A and B is a line segment
We can also calculate distance, indirectly based on two other measurements. Every line segment in the plane can be thought of as the hypotenuse of a right triangle (Figure 2).
Figure 2: is the hypotenuse of
Constructing such a triangle would then enable us to use the Pythagorean Theorem (Equation 1) to calculate distance, where a and b are legs, opposite of angles A and B, respectively and c is the hypotenuse, opposite the right angle, C.
To calculate the distance, we solve the Pythagorean theorem for c and substitute the values for our right triangle (Equation 2).
Sometimes, however it is inconvenient to measure distances directly, such as when the distances are very large. In such cases the indirect calculation of distance using the Pythagorean theorem would be helpful, except that to calculate one distance, we are now required to measure two additional distances. More convenient, still would be the ability to indirectly calculate the distance between two points, given only the locations of those points. Fortunately we can exactly locate any two planar points within the Cartesian-coordinate system (Figure 3).
Figure 3: in the Cartesian-coordinate system
We notice that each of the three points, A, B, and C have been assigned ordered pairs of the form (x, y), where x and y refer to the positive or negative distances of the points from 0 on the x-axis and y-axis, respectively. For instance, point A(-5.79, -1.56) is 5.79 cm left of zero on the x-axis and 1.56 cm below zero on the y-axis, whereas the point B(1.53, 2.80) is 1.53 cm right of zero on the x-axis and 2.80 cm above zero on the y-axis.
Working with these distances provides an easier approach to the measure of distance from A to B. Notice, also that b is equal to the difference between the x-ordinates of points A and B, which is approximately the Sketchpad measurement within rounding error (Equation 3).
Likewise, a is also equal to the difference between the y-ordinates of the points A and B, within rounding error of the Sketchpad measurement (Equation 4).
At this point we could substitute into the Pythagorean Theorem (Equation 1). However, we see that forand, the distances opposite of these vertices are given by Equations 5 and 6, respectively and so a more generalized approach may be taken.
Solving Equation 1 for c, we have Equation 7.
Now, substituting for a and b, from Equations 5 and 6, we have Equation 8.
Since this is a generalized equation for distance, we replace the c with d and make use of the commutative property of addition to write the distance formula, Equation 9.
Now, given points and , we calculate distance via substitution into the distance formula (Equation 10).
We have thus computed a one-dimensional measure of distance, the
length of a
line segment. Nevertheless, distance may also be measured within the
two- dimensions, such as the total distance around an enclosed region,
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