Height of Parallelogram or Trapezoid

Date: 04/30/99 at 00:27:59
From: Bob Underwood
Subject: Height in parallelograms and trapezoids

Could you explain the concept of height with regard to a parallelogram 
or a trapezoid?

Thanks,

Bob Underwood

Date: 04/30/99 at 08:39:53
From: Doctor Rick
Subject: Re: Height in parallelograms and trapezoids

Hi, Bob.

Each of these figures has a pair of parallel sides (the parallelogram 
has two such pairs). Pick one of these sides and call it the base; 
then the height is the distance between the line that contains the 
base and the line that contains the side parallel to it. 

The distance between two parallel lines is the length of a line 
segment with an end point on each of the lines and perpendicular 
to both lines. Any line in the plane that is perpendicular to one of 
the lines will be perpendicular to the other, and the length will be 
the same wherever the segment is located - the distance between 
parallel lines is constant along their length.

It may be that no line can be drawn perpendicular to the base such 
that one end lies on the base and the other on the parallel side. This 
is not a problem; we measure the distance between the two LINES, not 
the distance between the SEGMENTS of these lines that are the sides of 
the figure.

              .......__________________
              |     /                 /
              | /                 /
           /  | h              /
      /       |         /
  /___________|____/

The area of a parallelogram is the length of the base times the 
height. The area of a trapezoid is the mean of the lengths of the base 
and the side parallel to it, times the height:

            a
      _____________
     /      |      \
    /       |        \
   /        | h        \
  /_________|____________\
              b

b is the base, h is the height, a is the side parallel to the base.

In a parallelogram, you could choose any of the sides and call it the 
base, as long as you define the height perpendicular to this base; 
the area will be the same in any case:

               b1
        _________________
       /\  |            /
      /    |\          /
  b2 /     |  h2\     /
    /    h1|        \/ b2
   /       |        /
  /________|_______/
          b1

  Area = b1 * h1 = b2 * h2

If you had a specific question about height that I haven't answered, 
please write back.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/