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### Calculating Areas of Quadrilaterals and Triangles

```Date: 08/30/2005 at 18:22:18
From: Meghan
Subject: measurement

What portion of an acre is a plot of land 85 x 174 x 73 x 118?  What
portion of an acre is a plot of land 150 x 118 x 150?

I'm not sure of the dimensions of an acre.

```

```
Date: 08/30/2005 at 20:25:30
From: Doctor Rick
Subject: Re: measurement

Hi, Meghan.

This is perhaps the most common question we get that we cannot answer!

When you want to find the area of a quadrilateral (figure with four
straight sides), it isn't enough to know the lengths of the four
sides.  If you can imagine taking four sticks with lengths 85, 174,
73, and 118 inches, and hinging them together at the ends, you'll
find that the result is not a rigid figure.  You can change the angles
between the sides, and the area will change as you do that.

If, in addition to the lengths of the sides, you also knew the angle
between two of the sides, we could find the area.  (If you have a site
plan that shows not just the lengths but the directions of the sides,
we could find the angles between sides.)  Or if you could measure the
length of one of the diagonals, the distance between opposite corners
of the lot, that would be enough information too.

Now for the second question: if you're saying that this lot is a
triangle, the lengths of the sides *are* sufficient to find the area.
If you try the same hinged-stick experiment, you'll find that the
triangle is rigid.

The formula to use here is Heron's formula for the area of a triangle:

Area = sqrt(s(s-a)(s-b)(s-c))

where a, b, and c are the sides of the triangle, and

s = (a+b+c)/2

is the "semi-perimeter" (half the perimeter) of the triangle.  With

a = 150
b = 118
c = 150
s = (150+118+150)/2 = 209
Area = sqrt(209*59*91*59)
= sqrt(66205139)
= 8136.7

If the lengths are in feet, the area of the triangle is 8136.7 square
feet.  One acre is 43,560 square feet, so the area is

1 acre
8136.7 ft^2 * ----------- = 0.18679 acre
43,560 ft^2

If you can get a diagonal of the quadrilateral in your first question,
you can use this same method to find the area of each of the triangles
into which the diagonal divides the lot.  The area of the lot is the
sum of these two areas.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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