Geometry » Pythagoras' theorem
Pythagoras' theorem is also known as Pythagorean theorem.
You can do Pythagoras' theorem:
1. With a scheme/table2. With the shortest/fastest way
3. With an equation.
Directly to the examples
More theory
4. Pythagoras in space5. Reverse Pythagoras' theorem (to check whether the angle is 90°)
All three methods use a^{2} + b^{2} = c^{2}.
The theorem says that in a rightangled triangle the area of the two squares adjacent to the shorter sides added together is equal to the area of the area of the square adjacent to the longest side. In the figure below this means that the area of the two pink squares together is equal to the area of the blue square. The longest side is called the hypotenuse.
On the left, you can see that the pink area is the area of the whole square minus the four equal triangles. The area of the blue area is the same, also the area of the whole square minus the four equal triangles. On the right you can see those figures slid on top of each other in such a way that the yellow triangles are on top of each other. You now know that a^{2} + b^{2} = c^{2} must apply in the yellow triangle.
You can also say this:
In a rightangled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
Misleading name
Contrary to what you might think, it was not Pythagoras who was the first to came up with this theorem. This method was already known with the Sumerians and Babylonians (current Iraq) and also the Indians long before Pythagoras lived. The Greek and perhaps Pythagoras are probably the one who brought the theorem to the western world.
1. With a scheme/table
Method that is used in most modern mathematics books.
Often used to teach it to pupils who get it for the first time.
Below you can see the scheme that is used for this method.
side  square  
shorter side  
shorter side  +  
hypotenuse 
1.  In the left column you fill in the sides that are known/given. Put a question mark at the side you need to calculate. REMEMBER: The hypotenuse should always be at the bottom! 
2.^{ }  Calculate the squares of the known sides (#^{2}). 
3.  Calculate the square of the unknown side (by adding or subtracting). 
4.  Calculate the unknown side by using the . 
2. With the shortest/fastest way
Use Pythagoras' theorem in the following way:
See examples.
3. With an equation
This method is used in some maths books and is probably the method your parents learned. You really use a^{2} + b^{2} = c^{2} in that form.
You fill in the known sides directly in the formula and then start solving the equation.
See examples.
Examples
Example 1 Calculate the length of the side with the '?'. So you have to calculate the hypotenuse. 

Scheme/table  Shortest/fastest  Equation  
1. Fill in 16 and 30 2. Calculate 256 and 900 3. 256 + 900 = 1156 4. ? = = 34 
thus c = = 34 
a^{2} + b^{2} = c^{2} thus
 
Example 2 Calculate the length of the side with the '?'. You have to calculate a shorter side. 


Scheme/table  Shortest/fastest  Equation  
1. Fill in 3 and 7 2. Calculate 9 and 49 3. 49 – 9 = 40 4. PQ = 
thus PQ = = 
PQ^{2} + QR^{2} = PR^{2} thus


Example 3 Given are the points A(–10, 3) and B(5, 23). Calculate the distance between these two points. You have to calculate the hypotenuse. First you need to calculate the horizontal and vertical distance between A and B. A and B are horizontally 5 – –10 = 15 and vertically 23 – 3 = 20 units apart. In a sketch it will look like this:  
Scheme/table  Shortest/fastest  Equation  
1. Fill in 15 and 20 2. Calculate 225 and 400 3. 225 + 400 = 625 4. ? = = 25 
thus c = = 25 
a^{2} + b^{2} = c^{2} thus

4. Pythagoras in space
Within solids you often do not have any rightangled triangles. However, it is always possible to make your own, by using a crosssection. At the theory about long diagonals is an example of how you can calculate a long diagonal in a cuboid. It is also possible to use Pythagoras' theorem in other solids like pyramids and cones.
Example 1In cube ABCD.EFGH with edges measuring 4, point K is in the middle of BC.
Calculate EK.
To calculate EK you will have to use rightangled triangles.
In the cube you see two possible rightangled triangles you can use.
Using the green crosssection:
We can calculate EK with rightangled triangle EBK.
BK = 2, but BE is unknown.
BE is calculable using the front face.
Scheme/table  Shortest/fastest  
BE = 
BE = = 6  BE = = Now we can calculate EK. EK = = = 6 
Using the red crosssection:
We can calculate EK with rightangled triangle EKM.
KM = 4, but KM is still unknown.
EM is calculable using the top face.
Scheme/table  Shortest/fastest  
EM = 
BE = = 6  EM = = Now we can calculate EK. EK = = = 6 
Example 2
Pyramid ABCD.T has a square base with sides measuring 6 cm and slanted sides measuring 8 cm.
Calculate height ST.
To calculate ST you can use crosssection ACT.
Sketch crosssection ACT with height ST.
You can see that SCT will be the needed rightangled triangle.
AC and SC are already written in the sketch below, however you need to calculate them first.
Calculate, using the base, the length of AC first.
With AC you can calculate the length of SC.
AC = cm 
SC is half the length of AC. 
Now you can calculate ST in triangle SCT.
side  square  
SC = 12  18  
ST = ?  46  +  
AC = 8  64 
ST = ≈ 6,78 cm
5. Reverse Pythagoras' theorem
With the reverse of Pythagoras' theorem you can calculate/find out whether an angle in a triangle is greater than, less than or equal to 90°. With this method you always look at the angle opposite of the hypotenuse/longest side. In the newest maths books, this name might not be used. Although the method is explained.
It works like this:
Because a^{2} + b^{2} = c^{2} only works in a rightangled triangle you can use Pythagoras' theorem to check whether or not a triangle, of which you know all three sides, is rightangled or not. If not, you can also tell whether the triangle is acuteangled or obtuseangled.
The rules:
a^{2} + b^{2} = c^{2}. The triangle is rightangled.
a^{2} + b^{2} < c^{2}. The triangle is obtuseangled (c is too long and makes the angle bigger).
a^{2} + b^{2} > c^{2}. The triangle is acuteangled (c is too short and makes the angle smaller).
5^{2} + 12^{2} = 14^{2} 25 + 144 = 196 169 = 196 169 < 196 a^{2} + b^{2} < c^{2} So triangle ABC is obtuseangled. A > 90° 
16^{2} + 20^{2} = ()^{2} 256 + 400 = 656 656 = 656 a^{2} + b^{2} = c^{2} So triangle DEF is rightangled. A = 90° 
()^{2} + ()^{2} = 13^{2} 75 + 95 = 169 170 = 169 170 > 169 a^{2} + b^{2} > c^{2} So triangle GHI is acuteangled. A < 90° 