While there is no
simple formula to determine
area for most
quadrilaterals, and most
polygons for that matter, we saw
last
section that for the special quadrilaterals
parallelograms and
trapezoids there are specific formulas for
determining area. The area of a
triangle,
however, does. This is why it is so important that any polygon can be divided
into a number of triangles. The area of a polygon is equal to the sum of the
areas of all of the triangles within it.
The area of a triangle can be calculated in three ways. The most common
expression for the area of a triangle is one-half the product of the
base
and the height (1/2AH). The height is formally called the
altitude, and is
equal to the length of the
line segment with
one endpoint at a
vertex and the other endpoint on
the line that contains the
side opposite the
vertex. Like all
altitudes, this
segment must be
perpendicular to the line
containing the side. The side opposite a given
vertex is called the
base of a
triangle. Here are some triangles pictured with their
altitudes.
Another way to calculate the area of a triangle is called
Heron's Formula, named
after the mathematician who first proved the formula worked. It is useful only
if you know the lengths of the sides of a triangle. The formula makes use of
the term
semiperimeter. The semiperemeter of a triangle is equal to half
the sum of the lengths of the sides. Heron's Formula states that the area of a
triangle is equal to the
square root of s(s-a)(s-b)(s-c), where s is the
semiperimeter of the triangle, and a, b, and c are the lengths of the three
sides. The proof of Heron's Formula is rather complex, and won't be discussed
here, but his formula works like a charm, especially if all that is known about
a triangle is the lengths of its sides.
The third and final way to calculate the area of a triangle has to do with the
angles as well as the side lengths of the
triangle. Any triangle has three sides and three angles. These are known as
the six parts of a triangle. Let the lengths of the sides of the triangle equal
a, b, and c. If the
vertices opposite each length are angles of measure A, B,
and C, respectively, then the triangle would look like this:
Figure 5.2: A triangle with side lengths a, b, and c and angle measures A, B, and
C
The area of such a triangle is equal to one-half the product of the length of
two of the sides and the sine of their included angle. So in the triangle
above, Area=1/2(ab sin(C)), or Area=1/2(bc sin(A)), or Area=1/2(ac sin(B)).
The sine of an angle is a trigonometric property that you can learn more about
in the Trigonometry SparkNote.
With these formulas, it is possible to calculate the area of a triangle any time
you have any of the following information: 1) the length of the
base and its
altitude; 2) the lengths of all three sides; or 3) the length of two sides and
the measure of their included angle. With these tools, it is possible to
calculate the area of triangles, and, as we shall see, by summing the areas of
triangles within a polygon, it is possible to calculate the area of any polygon.
