We are all familiar with the formula for
the area of a triangle,
,
where b stands for the base and h stands for the height
drawn to that base. 

(the lettering used is of no importance) 
In the triangle at the right, the
area could be expressed as:

Now, let's be a bit more creative and
look at the diagram again. By using the right triangle on the left
side of the diagram, and
our knowledge of trigonometry, we can state that:
This tells us that the height,
h, can be expressed as bsinC.
If we substitute this new
expression for the height, we can write the
triangle area formula as:
(where a and b are adjacent
sides and C is the included angle) 

We have just discovered that the area of a triangle
can be expressed using the lengths of two sides and the sine of the
included angle. This is often referred to as the SAS Formula
for the area of a triangle.
The "letters" in the formula may change from problem
to problem, so try to remember the pattern of
"two sides and the sine of the included angle".
"Wow! A trig area formula for triangles!!!" 
We no longer have to rely on a
problem supplying us with the length of the altitude (height) of the
triangle in order for us to find the area of the triangle. If we know two sides and the included angle, we are
in business. 
Example 1:
Given the triangle at
the right, find its area. Express the area rounded to three
decimal places.

Be careful!!!
When using your graphing calculator, be
sure that you are in DEGREE Mode, or that you are
using the degree symbol. 



Example 2:

In Example 2,
we discovered, due to the doubling, that the
area of a parallelogram is really just
Parallelogram
(where a and
b are adjacent sides and C is the
included angle) 


Use the TI83+/84+
graphing calculator
with this new area formula.
Click here. 

