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!!
