Pythagoras' theorem states that the three sides of any rightangled triangle (a triangle that has a $90^\circ$90° angle), are related in a simple way.
Let's look at a rightangled triangle with sides measuring $a$a, $b$b and $c$c:
The side $c$c is the hypotenuse. The hypotenuse of a rightangled triangle can be readily identified because it is always:
Sides $a$a, $b$b and $c$c are related in the following way:
For any rightangled triangle:
"the square of the hypotenuse is equal to the sum of the squares of the two shorter sides"
$c^2=a^2+b^2$c2=a2+b2
where $c$c represents the length of the triangle's hypotenuse and $a$a and $b$b are the lengths of the two shorter sides.
Pythagoras' theorem can be represented by the following diagram, where the combined areas of the squares produced by the two shorter sides is equal to the area of the square produced by the hypotenuse.
If two sides of any rightangled triangle are known, and one side is unknown, Pythagoras' theorem can be used to find the length of the unknown side.
We can rearrange the equation to make the unknown side the subject:
If we want to solve for the hypotenuse:  $c=\sqrt{a^2+b^2}$c=√a2+b2 
If we want to solve for either of the two shorter sides: 
$b=\sqrt{c^2a^2}$b=√c2−a2 $a=\sqrt{c^2b^2}$a=√c2−b2 
A rightangled triangle has two shorter sides that are $10$10 cm and $12$12 cm in length. Find the length of the hypotenuse.
Think: We want to find the length of the longest side. The two shorter sides, $a$a and $b$b are given, so we can substitute directly into Pythagoras's theorem.
Do:
$c^2$c2  $=$=  $a^2+b^2$a2+b2 

$=$=  $10^2+12^2$102+122 
(Substitute $a=10$a=10 and $b=12$b=12) 

$=$=  $100+144$100+144 


$=$=  $244$244 


$c$c  $=$=  $\sqrt{244}$√244 
(Take the square root of both sides) 
$=$=  $15.6204$15.6204... 


$=$=  $15.62$15.62 cm ($2$2 d.p.) 
(Round to 2 decimal places) 
So, the hypotenuse is approximately $15.62$15.62 cm long.
If all we know about a triangle is the lengths of its sides, Pythagoras' theorem can be used to determine whether or not the triangle is rightangled.
For example, if a triangle has side lengths measuring $6$6 cm, $8$8 cm and $10$10 cm, we can substitute these values into Pythagoras' theorem. If the values satisfy the theorem, then the triangle must be rightangled.
The hypotenuse, $c$c, is clearly $10$10 cm, because it is the longest side. It doesn't matter how we assign $a$a and $b$b, but let's make $a=6$a=6 cm and $b=8$b=8 cm.
$c^2$c2  $=$=  $10^2$102 
$=$=  $100$100 
$a^2+b^2$a2+b2  $=$=  $6^2+8^2$62+82 
$=$=  $36+64$36+64  
$=$=  $100$100 
Since $c^2=a^2+b^2$c2=a2+b2, a triangle with sidelengths $6$6, $8$8 and $10$10 units, must be a rightangled triangle.
Calculate the value of $c$c in the triangle below.
Calculate the value of $b$b in the triangle shown. Give your answer correct to $2$2 decimal places.
Find the length of the unknown side $b$b in a rightangled triangle whose hypotenuse measures $6$6 mm and one other side measures $4$4 mm.
Give your answer correct to two decimal places.
Find the length of the unknown side, $x$x, in the given trapezium.
Give your answer correct to two decimal places.