The Mathematics 11
Competency Test
The Theorem of Pythagoras With
Applications
The so-called Theorem of Pythagoras states that for
a right triangle, it is always true that
A
c2 = a2 + b2
Recall that in a right triangle, the longest side,
opposite the right angle, is called the hypotenuse.
Thus the Theorem of Pythagoras states that for right
triangles
b
C
c
B
a
the square of the length of the hypotenuse
= the sum of the squares of the lengths of the other two (shorter) sides
If the lengths of any two sides of a right triangle are known, then the length of the remaining third
side can be computed using this formula.
(In the examples that follow, we will observe the convention that for right triangles, the right angle
is always labelled C, and the other two angles are labelled A and B. The labels for the sides are
the lower-case characters of the angles opposite. So, the side opposite the right angle vertex is
always labelled c, the side opposite vertex A is always labelled a, and the side opposite vertex B
is always labelled b. These symbols are somewhat loosely used simply as names of these parts
of the triangle, and as well, in formulas, they represent the sizes of the parts. Thus b is a label for
the side opposite vertex B, but in a formula, the symbol b stands for the length of that side.
Similarly, the symbol B is a label for a vertex, but also, in formulas represents the size of the
angle at the vertex, in some appropriate unit of measurement such as degrees.)
Example 1: The two perpendicular sides of a right triangle have lengths 12.3 m and 8.7 m,
respectively. Compute the length of the hypotenuse of this right triangle.
solution:
The situation is shown in the sketch to the
right. We are asked to determine the
value of c. Applying Pythagoras’s
Theorem directly gives
b = 8.7 m
c
c2 = a2 + b2
= (12.3 m)2 + (8.7 m)2
a = 12.3 m
= 226.98 m2
Therefore,
c
c2 
226.98 m 2  15.0659 m
Since the original measurements are rounded to one decimal place, we should probably do so
with our answer. Thus, the length of the hypotenuse of this right triangle is 15.1 m.
David W. Sabo (2003)
The Theorem of Pythagoras With Applications
Page 1 of 4
Example 2: A plot of land in the shape of a right triangle has a hypotenuse of length 438 m, and
one of the other sides is 263 m long. Determine the area in hectares of this plot of land. (Hint:
1 hectare = (100 m)2)
solution:
The situation is sketched in the figure to the right. Here
we have decided to associate the shorter side 263 m
long with the symbol “a”.
Now, the goal is to compute the area of this triangular
region. With a right triangle, the two perpendicular sides
can be considered to form a base and the corresponding
altitude of the triangle. So, for this specific triangle, we
have that
Area 
c = 438 m
b
a = 263 m
1
1
base  altitude  ab
2
2
Thus, to calculate the area, we need to calculate the length ‘b’ of the side oriented vertically in the
figure. But, by Pythagoras’s Theorem
c2 = a2 + b2
we can write
b2 = c2 – a2
= (438 m)2 – (263 m)2
= 122675 m2
Thus
b
b2 
122675 m 2  350.25 m
Now we can calculate the area:
Area 
1
1
ab   263 m  350.25 m   46057.875 m 2
2
2
Now we need to convert this answer to hectares. Since
1 hectare = (100 m)2
we get
Area  46507.875 m 2 
1 hectare
100 m 
2
 4.6057875 hectares
Since the original measurements were rounded to a whole number of meters, our final answer
should be rounded to the equivalent of a whole number of square meters, which amounts to
rounding the final answer in hectares to the equivalent of 46058 m2. Thus, our final answer is
that the area of the triangular region is 4.6058 hectares.
David W. Sabo (2003)
The Theorem of Pythagoras With Applications
Page 2 of 4
Example 3: Hank wishes to determine the height of his roof gutters above the ground. He
doesn’t have a ladder and the gutters are too high for him to reach so that he can measure their
height directly. However, he has a straight piece of lumber which is exactly 16 feet long. When
he places it so that one end just touches the roof gutter, the bottom end is 4 feet 7-3/4 inches
from the point on the ground directly below the gutters. Determine the height of the gutters above
the ground at this location.
solution:
We need to assume that the ground is horizontal in the
vicinity of the vertical house walls. Then the overall
situation is as shown in the sketch to the right. Hank wishes
to determine the distance h. We see that the 16 foot long
piece of lumber forms the hypotenuse of a right triangle,
which has as its other two sides the distance of 4 feet 7.75
inches that Hank was able to measure along the ground
between the bottom of the piece of lumber and the point
vertically beneath the edge of the gutters, and the unknown
but desired height of the roof gutters above the ground.
Since the unknown height is the third side of a right triangle
for which we know the lengths of the other two sides, we
can use Pythagoras’s Theorem to solve this problem.
h
4 ft 7.75 in
In order to do arithmetic with these lengths, they must all have the same
units. It is most convenient to use units of feet here. The length of the
hypotenuse of the right triangle is already in these units. However, the
distance along the ground is given in mixed units of feet and inches.
Converting to feet, we get
h
3
1 ft
4 ft  7 in  4 ft  7.75 in 
 4.6458 ft
4
12 in
rounded to four decimal places.
So, now, by Pythagoras’s Theorem,
4.6458 ft
h2 = (16 ft)2 – (4.6458 ft)2
 234.4165 ft2
and so
h  234.4165 ft 2  15.31 ft
Thus, we conclude that the gutters around Hank’s roof are 15.31 feet above the ground.
Caution: People are sometimes tempted to apply Pythagoras’s Theorem even when the triangle
in question is either not a right triangle or is not known for sure to be a right triangle.
Pythagoras’s Theorem only works for right triangles. If it is applied in any way to triangles which
are not right triangles, it is sure to give erroneous results.
David W. Sabo (2003)
The Theorem of Pythagoras With Applications
Page 3 of 4
If we know the lengths of the three sides of a triangle, Pythagoras’s Theorem can also be used to
determine whether or not that triangle is a right triangle.
Example 4: Each of the following sets of three numbers are claimed to be lengths of sides of
right triangles. Confirm that this is true in each case.
(i) 8, 19, 13
(ii) 13, 12, 5
(iii) 7, 42, 9
solution:
In each case, the longest side must be the hypotenuse (since the hypotenuse is always the
longest side of a right triangle). If the given triangle is really a right triangle, then the square of
the length of the longest side must be exactly equal to the sum of the squares of the lengths of
the other two sides.
So, for case (i), the length 19 must be the length of the hypotenuse. Here
192 = 361.
But
82 + 132 = 64 + 169 = 233.
Since 233  361, these three numbers do not satisfy Pythagoras’s Theorem, and so they cannot
represent the lengths of the three sides of a right triangle.
For case (ii), the length 13 must be the length of the
hypotenuse. Then
13
132 = 169
5
and
52 + 122 = 25 + 144 = 169
Since these two results are equal, we conclude that these three
lengths do form a right triangle, as sketched in the figure.
12
For case (iii), the length 42 would have to be the length of the hypotenuse. Then, checking, we
find that
422 = 1764
and
72 + 92 = 49 + 81 = 130
Since 1764  130, we know that these three lengths cannot form a right triangle. In fact, since 42
is greater than 7 + 9 = 16, it turns out that these three lengths cannot even form a triangle of any
shape!
David W. Sabo (2003)
The Theorem of Pythagoras With Applications
Page 4 of 4