Pythagoras’ Theorem
Use the right angle to choose the hypotenuse, then find a missing length with confidence.
The one condition
Pythagoras’ theorem works only in a right-angled triangle. The hypotenuse is the side opposite the right angle and is always the longest side.
$a^2+b^2=c^2$
Finding the hypotenuse
Example: both shorter sides are known
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
Step 1 Substitute into $a^2+b^2=c^2$.
$6^2+8^2=c^2$
Step 2 Add the squares.
$36+64=100$, so $c^2=100$.
Step 3 Take the positive square root.
$c=[[sqrt]]{100}=10$ cm.
Finding a shorter side
Example: the hypotenuse is known
The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other shorter side, $x$.
Start with the hypotenuse squared, then subtract the known shorter side:
$x^2=13^2-5^2=169-25=144$
$x=[[sqrt]]{144}=12$ cm.
Avoid the common mix-up
Add the two squares only when finding the hypotenuse. When finding a shorter side, subtract the known shorter side’s square from the hypotenuse’s square. Never subtract from a shorter side.
Coordinates use the same triangle
For points $A(x_1,y_1)$ and $B(x_2,y_2)$, the horizontal and vertical changes form the shorter sides.
$AB=[[sqrt]]{(x_2-x_1)^2+(y_2-y_1)^2}$
For $A(1,2)$ and $B(4,6)$, the changes are 3 and 4, so $AB=[[sqrt]]{3^2+4^2}=5$.
Final check
Your hypotenuse answer must be longer than either shorter side. Keep full calculator values during working and round only at the end if a question asks you to.
Practice Problems
Sketch or mark the right angle before calculating.
Practice 1
A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.
Show answer
$c^2=9^2+12^2=81+144=225$
$c=15$ cm.
Practice 2
The hypotenuse is 17 m and one shorter side is 8 m. Find the other shorter side.
Show answer
$x^2=17^2-8^2=289-64=225$
$x=15$ m.
Practice 3
Find the distance between $A(-2,1)$ and $B(3,13)$.
Show answer
The horizontal change is 5 and the vertical change is 12.
$AB=[[sqrt]]{5^2+12^2}=[[sqrt]]{169}=13$.