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!!
