Nipissing University
Mathematics Lecture Talks
presents
The Area of a Triangle
It Is More Complicated
Than You Might Think!
Area is a physical quantity expressing the
size of a part of a surface.
Surface area is the summation of the areas
of the exposed sides of an object.
Square units (e.g. cm2) are used in
quantifying the measures areas and
surface areas.
The area of a geometric plane figure such as
a polygon is the measure of the number of
square units the object or plane figure is
made up of.
A triangle is a polygon with three vertices and
three sides which are straight line segments.
A polygon is a closed planar path composed of a finite
number of sequential line segments. The straight line
segments that make up the polygon are called its
sides or edges and the points where the sides meet
are the polygon's vertices. If a polygon is simple, then
its sides (and vertices) constitute the boundary of a
polygonal region, and the term polygon sometimes
also describes the interior of the polygonal region (the
open area that this path encloses) or the union of both
the region and its boundary.
The Vertices
A
The sides
C
B
The area
The Triangle
Computing the area of a triangle
Using geometry
The area S of a triangle is S = ½bh, where b is the
length of any side of the triangle (the base) and h
(the altitude) is the perpendicular distance between
the base and the vertex not on the base. This can
be shown with the following geometric construction.
Applications of Mathematics
Some Trivial
Applications:
• handle your money
• tell time
• gamble
• construct objects
• paint things
Some Exciting • solve problems
Applications: • discover new relationships
• create new formulas
Speed Away From Us
What is
the
scientist
(Hubble)
trying to
convey
to us
here?
Galaxies In
the Universe
The
Big
Bang
Distance Away from Us
Edwin Hubble’s graph
What is the artist
(Pablo Picasso)
trying to convey
in these
paintings?
What meaning is the mathematician
trying to covey from this odd-looking
expression?
Actually it represents a new
formula for finding the area
of a triangle when its vertices
x1
y1
on the Cartesian plane are
x2
y2
known!
x3
y3
x1
y1
Add the “down
products” and subtract
the “up products” and
take ½ of this result.
The formula can be extended to finding the area
of any convex polygon on the Cartesian plane:
A
1
2
x1
y1
Remember that the points must
be in counterclockwise order.
x2
y2
 12 x1 y2  x2 y1  x2 y3  x3 y2  ...
x3
y3



xn
yn
x1
y1
...  xn1 yn  xn yn1  xn y1  x1 yn 
Add the “down products”
and subtract the “up
products”.
In geometry, two sets are called congruent if one can be
transformed into the other by an isometry, (i.e., a
combination of translations, rotations and reflections). In
less formal language, two sets are congruent if they have
the same shape and size, but are in different positions (for
instance one may be rotated, flipped, or simply moved).
Congruent triangles
are triangles with the
same shape and size
and thus must have the
same area.
C
A
B
F
E
We write: ABC  DEF
D
There are actually four conditions under which
triangles can be proven to be congruent. Each
requires that three parts of the triangle (angles or
sides) be respectively equal in each triangle.
RHS (you are given a right angle, the hypotenuse
and one other side)
SAS (you are given two sides and the contained
angle)
SSS (you are given three sides)
AAS (you are given two angles and any side)
There is also an interesting fifth case: you are given
two sides and the angle is not contained. Two
possible triangles exist here (the ASS case).
The SAS case (the given angle
must be contained between the
given two sides):
Given C
C
and sides “a”
and “b”,
a find side “c”.
b
B
A
c
• here we use the cosine law
Now
let’s
find
its
area
The SSS case:
Given sides “a”,
“b” and “c”
C
a
b
find one of the
angles (C in
this case).
B
A
c
• here we use the cosine law
Now
let’s
find
its
area
The AAS case (the given side does not
have to be contained – just find the 3rd
angle by subtracting from 180):
Given A
and C
C
and side “c”,
a
b
find side “a”.
B
A
c
• here we use the sine law
Now
let’s
find
its
area
The ASS case (the given angle is not
contained between the given two sides):
Given A
C
a
b
and sides “a”
and “b”,
find angle B.
A
c
• here we use the sine law and
two triangles may be found
B
Two
Triangles
a<b
C
b
a
a
c
A
c
B
B
We use the sine law for each of the ASS cases.
One Triangle
C
a>b
a
a
b
B
B
c
A
c
One Triangle (special case)
when a = bsinA
B = 90
A
C
b
c
Now let’s find their areas
a
B
Summary of the area formulas:
1
A

bh
The RHS case:
2
A  1 ab sin A
2
The SAS case:
The SSS case:
A  s( s  a)( s  b)( s  c)
The AAS case:
abc
where s 
2
2
a tan B tan C
A
2(tan B  tan C )
2
2
2
2
The ASS
b
sin
A
b
cos
A

a

b
2
A  b sin 2 A 
case:
2
Thank you
for your
attention
and
participation