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G.GMD.1 STUDENT NOTES WS #3
1
THE TRIANGLE
The triangle area formula also relies heavily on the area of a rectangle. There are a number of different ways
to conceptually understand the area formula for a triangle. We will look at a few of them. Each method will
not only help us work with triangles but with many other polygons as well.
Formulas are derived by how one perceives the situation. There are often two approaches to this in area… Do
you see the shape as a shape within a larger shape or do you see it as a shape made up of many smaller
pieces. In other words, do you see the shape as a piece of a whole or do you see it as the whole with pieces
within it? First we will look at the triangle as a piece of a larger whole and the larger whole will be a rectangle.
TRIANGLE AREA – HALF OF THE WHOLE
height
base
base
1
1
height
height
height
height
1
base
2
1
height
2
base
base
base
Rectangle A = bh
1
Triangle A = bh
2
Rectangle A = bh
1
Triangle A = bh
2
Rectangle A = bh
1
Triangle A = bh
2
In the first two examples we can easily see how a triangle is exactly half the area of the rectangle that has the
same base and height as it. In the third example, it isn’t quite as easy because the rectangle is not the one
that encloses it as it was in the first two examples. The rectangle that encloses the third example is NOT the
rectangle that the triangle is half the area of.
Diagram #1
Diagram #2
height
height
base
x
base
It is hard for a student to understand why the rectangle in diagram #2 is not the correct rectangle and because
of the shape of the triangle is hard to prove or disprove it by dissection. This is another situation when
shearing helps us greatly visualize the correct relationship. If we fix the base and then move the vertex
opposite the fixed base on a parallel line the area remains fixed because the base and height do not change.
By shearing the triangle we can inscribe the triangle in the rectangle that has the same height and base, thus
again establishing that Area of a triangle is ½ (base)(height).
1
height
height
base
1
1
height
2
1
2
base
base
All of these triangles are have the same area because they have the same height and base and all of them are
exactly half the area of the rectangle with the same base and height, thus Area = ½ bh.
AREATRIANGLE = 1 bh
2
G.GMD.1 STUDENT NOTES WS #3
2
TRIANGLE AREA – DISSECTION
Another common technique for determining area is to cut the shape up into smaller pieces and then to
reorganize them into a shape that is easier to work with. This is a great method for area because each piece
maintains its original area value and so whatever the new shape is it will have the same area as the original
shape. So we are going to take a triangle and see if we can form a rectangle from it and then see what the
dimensions of the rectangle are compared to our original triangle.
height
height
height
base
1
2
height
base
base
base
Notice that by cutting the height in half we create a triangular piece
that we can rotate 180° to form a rectangle. The rectangle that is
formed has the same area as the original triangle because we are using
the pieces from the original triangle. The dimensions of the equivalent
rectangle are ½ the height and the same base. Thus Area = ½ bh.
Let us try this again with an oblique triangle to verify that this can work with any triangle.
h
h
h
base
base
base
Dissection helps us to change the triangle into a rectangle. The rectangle
that is formed has dimensions of ½ the height of the original triangle and
the same base. Thus Area = ½ (height)(base) = ½ bh.
1
2
h
base
This same process can be used to create a rectangle with the same area but with different dimensions than
the above two examples. This is possible because of the commutative property of multiplication, (½
height)(base) = (height)(½ base).
heght
heght
base
heght
base
base
This time we are halving the base to form our rectangle. The dimensions
of this rectangle differ from the above examples but the area is still the
same. This again establishes that the area of triangle is again ½ bh.
heght
1
2
base
1
1 
1 
h
b
=
b
h
=
bh = ATriangle




2
2 
2 
G.GMD.1 STUDENT NOTES WS #3
3
Given the following triangles, calculate their areas.
30°
4 cm
10 cm
8 cm
10 cm
12 cm
9 cm
Area = _____________
A = ½ bh
A = ½ (9)(4) = 18 cm2
Area = _____________
102 = 82 + x2
x=6
Area = _____________
Short leg = 6 cm
Long leg = 6 3
A = ½ bh
A = ½ (12)(8) = 96 cm2
A = ½ bh
A = ½ (6)( 6 3 ) = 18 3 cm2
Dissect this triangle to determine its area.
Area = 2(5) = 10 cm2
Shear this triangle to determine its area.
Area = ½ (4)(1) = 2 cm2
This could also be done using a box technique
3
Area of Rectangle = 2(3) = 6 cm2
Area of Triangle 1 = ½ (2)(2) = 2 cm2
Area of Triangle 2 = ½ (1)(3) = 1.5 cm2
Area of Triangle 3 = ½ (1)(1) = 0.5 cm2
1
2
Area = 6 – 2 – 1.5 – 0.5 = 2 cm2
G.GMD.1 WORKSHEET #3
NAME: ____________________________ Period _______
1
1. Create/Draw a rectangle that is exactly double the area of the triangle provided.
a)
b)
c)
d)
e)
f)
2. Victoria says that there is more than one way to draw the rectangle that is exactly double the given
triangle. Find three different rectangles.
a)
b)
c)
3. Draw in the height for the given base.
a) base is BC
b) base is AC
B
c) base is BC
A
B
A
d) base is AB
A
A
B
B
C
C
C
C
4. Explain how the rectangle area is exactly double the area of the triangle.
G.GMD.1 WORKSHEET #3
2
5. Using dissection, demonstrate why the area formula for a triangle is A = ½ bh. Label the dimensions of
the resulting rectangle in relationship to the original triangle.
a)
b)
c)
h
h
h
base
base
base
6. Find the area triangle.
a)
b)
Area = ______________
Area = ______________
7. Determine the area of the following triangles.
a)
b)
c)
5 cm
12 cm
6 cm
6 cm
5cm
12 cm
10 cm
10 cm
Area = ____________
d)
3 cm
Area = ____________
e)
6 cm
2 cm
Area = ____________
f)
10 cm
5 cm
8 cm
5 cm
Area = ____________
Area = ____________
Area = ____________
G.GMD.1 WORKSHEET #3
3
8. Determine the area of the following triangles.
a)
b)
c)
6 cm
60°
13 cm
8 cm
10 cm
60°
60°
10 cm
Area = ____________
Area = ____________
Area = _________________ (E)
d)
e)
f)
8 3 cm
11 cm
6 cm
65°
30°
39°
15 cm
4 cm
Area = ____________ (2 dec.)
Area = ____________ (2 dec.)
Area = _________________ (E)
g)
h)
i)
16 cm
60°
45°
12 cm
4 cm
Area = ________________ (E)
Area = ____________
9 cm
Area = ___________________ (E)
G.GMD.I
WORKSHEET #3
1. Create/Draw a rectangle
NAME:
Period
that is exactly double the area of the triangle provided.
b)
a)
c)
f)
2. Victoria says that there is more than one way to draw the rectangle that
triangle. Find three different rectangles.
lr{-1
a)
b)
is exactly double the given
c)
3. Draw in the height for the given base.
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a) base is
BC
b) base is AC
c) base is BC
A'
CC
4. Explain how the rectangle area is exactly double the area of the triangle.
d) base is AB
A
G.GMD.I
2
WORKSHEET #3
5. Using dissection, demonstrate why the area formula for a triangle is A
the resulting rectangle in relationship to the original triangle.
=lrbh.
Labelthe dimensions of
c)
4t\
7 L \
--l
/1
h
T
\
\
bi ;e
6. Find the area triangle.
b)
a)
\
Area
= l"r
\\
\
ai,tt-
Area =
7. Determine the area of the following triangles.
a)
b)
,.rN
r!,)
2
12 cm
10 cm
Area
d)
= 3o
rye)
ctY,u
Area
f)
Area =
e)
=
3
8cm
2--
Area =
Area =
jo
eato,>
Area =
VO Cfit&-
G.GMD.I
WORKSHEET #3
8. Determine the area of the following triangles.
T:;;:';,',
lW
a)
b)
c)
huo
h=la
7t\
6p"
13 cm
(
I
I @
k2
10 cm
k= *)
Area
1.
= bo CAL
3L
Area =
d)
cilAL
(E)
f)
15 cm
$rw
31. *
117tn7t.la.=6'1L
ft=t@
(2 dec.)
Area =
h)
Ir :
Area
=
Area =
(E)
i)
h,=,r(q)
S@rt)
2-
h---
q (t+ttn)
3z+ SZE cn" 1g1
aec.)
4cm
#^E
ft=
cyLiz
<-u\;.*-
2, L
U"-- b
Area =
'" /r,r)ut-\
N--
Area= V1 crrtL
6(v)
(E)
)