# Pythagoras’s Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

i.e.: c² = a² + b² in the following diagram:

image: http://revisionworld.com/sites/revisionworld.com/files/imce/trig3.gif

**Example**

Find AC in the diagram below.

AB² + AC² = BC²

AC² = BC² – AB²

= 13² – 5²

= 169 – 25 = 144

AC = 12cm

image: http://revisionworld.com/sites/revisionworld.com/files/imce/Pythag.gif

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# Pythagoras’s Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

i.e.: c² = a² + b² in the following diagram:

image: http://revisionworld.com/sites/revisionworld.com/files/imce/trig3.gif

**Example**

Find AC in the diagram below.

AB² + AC² = BC²

AC² = BC² – AB²

= 13² – 5²

= 169 – 25 = 144

AC = 12cm

image: http://revisionworld.com/sites/revisionworld.com/files/imce/Pythag.gif

This video explains how to work out phytagoras theorem

**3D Problems (Higher Tiers)**

In higher tier papers, you may be asked to solve 3d problems using Pythagoras.

**Example**

A cuboid has sides of length 10cm, 2 √11 cm and 5cm. Find the length of a diagonal.

image: http://revisionworld.com/sites/revisionworld.com/files/imce/py3d1.GIF

So we want to find AD.

We can draw a right-angled triangle inside the cuboid which has AD as it’s hypotenuse. Then we can use Pythagoras.

image: http://revisionworld.com/sites/revisionworld.com/files/imce/py3d2.GIF

To use Pythagoras, we need to know AC and CD. We know that CD is 5 cm. We need to find AC.

We can use Pythagoras to find AC, because if we look at the cuboid from above, we see that AC is the diagonal of a rectangle

image: http://revisionworld.com/sites/revisionworld.com/files/imce/py3d3.GIF

ABC is a right angled triangle, so by Pythagoras, AC^{2} = AB^{2} + BC^{2}

= 10^{2} + (2 √11)^{2} = 100 + 44 = 144

Now we can find AD: AD^{2} = AC^{2} + CD^{2} = 144 + 25 = 169

Therefore AD = 13