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Pythagoras’ Theorem
In relation to Mathematics, Pythagoras’ theorem is the relation of the three sides of a rightangled triangle. The theorem can be expressed as the equation below, and relates to the
lengths of the sides of the triangle.
c2=a2+b2
c
a
“a” & “b” are the 2 shorter
sides and “c” is the
Hypotenuse (longest side)
b
Pythagoras’ theorem can also be rearranged to find a shorter side. For example:
a2 = c2-b2
Example 1:
or
b2 = c2-a2
For the triangle below, calculate the length of the Hypotenuse, x, correct to 1
decimal place.
Step 1: Label the sides: a=4, b=7, c=x
x
4
Step 2: Substitute the values in Pythagoras’
Theorem:
c2 = a2 + b2
x2 =4 2 + 72
x2=16 + 49
x = 65
= 8.1
Example 2:
7
You can also use Pythagoras’ Theorem to work out if a triangle has a right
angle
Step 1: Does c2 = a2 + b2?
25
24
Step 2: 252 = 72+242 = 625 = 49 + 576
Step 3: 252 = 625
7
Both sides of the equation are equal so it is definitely a right angle triangle!
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Pythagorean Triads
On certain occasions, all 3 sides of a right angled triangle will be whole numbers. This is
called a Pythagorean Triad (also called a Pythagorean Triple). The right angled triangle below
is an example of a Pythagorean Triad.
5
4
3
We can use the following formula to create a Pythagorean Triad. Firstly, we need to find the
middle number “m” of a Pythagorean Triad, where “s” is an odd number. The third number
can then be found using Pythagoras Theorum.
m=
s2 −1
2
Example 3: How to create a Pythagorean Triad
Step 1: Choose any odd number for your “s” value and then square it. We will
choose “72 = 49” for this example.
Step 2: Find 2 consecutive numbers that add up to the squared value in step 1.
24 + 25 = 49
If you have trouble doing this in your head, then use the formula mentioned
above.
72 −1
Enter 7 into the equation: m =
2
Step 3: We end up with m = 24.
Step 4: Write down the number you squared in step 1 and the 2 numbers from step 2.
You now have your triple.
7, 24, 25
You can use Pythagoras to check the result.
c2=a2+b2
252=242 + 72
625 = 576 + 49
25
7
24
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