9 - 2 Area of a Triangle
 
 
 
 The formula to find the area of a triangle is K = bh/2 where K is the area, b is the base and h is the height which is perpendicular to the base.  How do we find the area if we don't have the height?
 
 
Given / B, a, and c, how can we find the area?
By dropping down a perpendicular from A we note by trig that:
sin B = h/c
c Sin B = h
substituting into the area formula we have:
K = (ac Sin B)/2
We could be given any two sides and the included angle and find the area.  The other forms are:
K = (bc Sin A)/2
K = (ab Sin C)/2
In words, this means:
K = (1/2) . (one side) . (another side) . (sine of the included angle)

 
Sample Problems
 
1)  Two sides of a triangle have lengths 8 m and 5 m.  The included angle measures 53o.  Find the area.
Solution:
K = (1/2)(8)(5)Sin 53
K = 20 Sin 53
K = 16.0 m2
 
2)  The area of a triangle is 20.  If a = 6 and b = 12, find all possible measures of / C.
 
Solution:
20 = (1/2)(6)(12) Sin C
20 = 36 Sin C
20/36 = Sin C
Sin-1(20/36) = C
33.7o = C
Since the angle could be acute or obtuse, there is another answer.
180 - 33.7 = 146.3o
C = 33.7o or 146.3o
 
3)  Find the area of the following quadrilateral:
 
Solution:
To find the area, we must split the figure into two triangles.  One way is as follows:
 
To find area A:
This is a right triangle, so one side is the base, the other the height.
K = (1/2)(5)(8)
K = 20
The area of A is 20 square units.
 
To find the area of B:
We need to find the length of the diagonal (in blue) and the size of the angle included between the blue length and length of 14.
We can find the length by using pythagorean's theorem.
                                                                                _______
= \/ 25 + 64
= 9.4
 
To find the angle, we need to first find the angle included between the sides 8 and 9.4 in the right triangle.  Using trig we get:
tan x = 5/8
x = tan-1(5/8)
x = 32o
Since we know the big angle is 75o, all we have to do is subtract to find the angle we need!
75 - 32 = 43
 
Now we have the included angle in triangle B.
K = (1/2)(14)(9.4) Sin 43
K = 44.9
To find the area of the quadrilateral, simply add the two areas:
44.9 + 20 = 64.9
The area is 64.9 square units.

We can now head on to the Law of Sines!!