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Topic: [math-learn] Stem & Leaf Plots with Decimals
Replies: 21   Last Post: May 5, 2008 5:31 PM

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 Ladnor Geissinger Posts: 313 From: University of North Carolina at Chapel Hill Registered: 12/4/04
[math-learn] length and area; sidewalks and roads
Posted: May 5, 2008 12:45 AM
 att1.html (14.9 K)

In response to several recent email notes about computing areas of
simple geometric regions, I will resend my note to math-learn from Nov
2003. That original note was in response to comments by Rex Boggs. I
hope he forgives me for throwing it out there again.
================================================================
circumference and area and remembering formulas and knowing which to
use when. Perhaps some teachers would find it useful to develop the
idea of computing area from two common examples of geometric regions:
constant width sidewalks (and borders and ribbon paths) and roads.
For each of these the computation of area is simple: multiply the
constant width w by the centerline length m.

Note that for any trapezoid with width w (distance between
parallel sides) and mid-line length m, the area is w*m. And borders
around parts of buildings, patios, and other objects are often made up
of a collection of such trapezoids all of the same width w and with
two successive trapezoids meeting at a corner along a line which makes
the same angle with each edge at the corner. Sidewalks are usually
made this way. So the total area of such a sidewalk or border is the
product of the constant width w multiplied by the sum of the mid-line
lengths of each of its segments.
Also, suppose we take a long piece of ribbon and tape a string to its
center-line and then start pasting in down flat onto a
surface. When we want to have it go off in a different direction we
cut the ribbon but not the string and flip the ribbon over and
continue with the two cut edges realigned. Clearly the total area of
our ribbon path is the width multiplied by the total mid-line length
of the string (no matter at what angle the ends are cut). Of course a
special case is a single triangle with w the "altitude" from vertex C
dropped to side AB (possibly extended), and the area is w*m where m is
the length of the midline parallel to AB and halfway between AB and C.
So our "formula" works for rectangles, squares, triangles,
trapezoids,..-- sidewalks and pieces of them.

Roads of constant width w are nearly a series of rectangles of width w
connected by sectors of circular rings. In this case again, the total
surface area of such a road is the product of w and the center-line
length. Does this kind of area computation appear in any texts? Note
that special cases include a sector of a ring or the whole ring for a
circular race track, and the really special case of a sector of a
circle or a whole circle. Note that one could argue that this works
for roads by approximating closely a road by using a ribbon path.

On the other hand, if we do know a formula for the area of a circle,
we could consider two concentric circles with radii a>b. Then for any
sector with angular measure t, the area of the part of the ring
between the two circles and within the sector is
(t/2pi)*pi*a^2 - (t/2pi)*pi*b^2 or
(a-b)*t*(a+b)/2. And a-b is the constant width of that ring while
t*(a+b)/2 is the length of the midline, that is, the arc of the circle
of radius (a+b)/2 which lies in this sector. So our simple formula for
area has appeared again.

Date Subject Author
4/9/08 Jon Wilson
4/10/08 Jon Wilson
4/10/08 Prof Martin Weissman
4/10/08 Ron Ferguson
4/10/08 Prof Martin Weissman
5/4/08 lolly45101961
5/4/08 Ed Wall
5/4/08 Marie Bahlert
5/4/08 Rex Boggs
5/4/08 Ed Wall
5/5/08 GS Chandy
5/5/08 Rex Boggs
5/5/08 Ralph A. Raimi