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Using Pythagoras' Theorem on 3D Shapes

In this worksheet, students will revise Pythagoras' Theorem and learn how to use it in 3D shapes.

'Using Pythagoras' Theorem on 3D Shapes' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR,

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation Triangle Mensuration

Difficulty level:  

Worksheet Overview

Pythagoras (top bloke), due to his love of mathematics and science was treated as a god in his time!

Did you know that Pythagoras was in fact responsible for the musical scales we have today?




He had many followers who hung on to his every word, they thought his theories were magical.


Let's remind ourselves of his theory:



To find the Hypotenuse of a triangle


1. Square one side 2² = 4²

2. Square the other side 6² = 36²

3.  Add your answers together 36 + 4 = 40² 

4. √40 = 6.32 to 2 decimal places


To find a shorter side




1. Square the number on the hypotenuse 16 x 16 = 256 cm²

2. Square the number of the shorter side given 11 x 11 = 121 cm²

3. Subtract your answers 256 - 121= 135 cm²

4. Square root your answer √135 = 11.62 to 2 decimal places


Now Pythagoras (who ate, slept and drank mathematics) didn't want to end it there, oh no!


He showed everyone how to apply this to 3D shapes. There is no end to his talent...


Calculating is just the same as normal when working with a 2-dimensional triangle.

The trick here is being able to spot the triangle as shown in the example below.

To start

1. Draw the line BE onto the shape

2. Then draw in the BC and EC to form a right-angled triangle.


To find the length of BE

1. You need the length of BC which is given as 5 cm

2. You need the length CE which is not given


boy looking glim


Don't look so glum.  Pythagoras has given us some information to help us!


You can see that there is another right-angled triangle given between C, E and F. We can use this to find the length CE


1. Square one side 5² = 25

2. Square the other side 5² = 25

3.  Add your answers together 25 + 25 = 50² 

4. √50 = 7.07 to 2 decimal places


Now we are ready to rock and roll

1. Square BC 5² = 25

2. Square EC 7.07² = 49.98

3.  Add your answers together 25 + 49.98 = 74.98² 

4. √74.98 = 8.66 to 2 decimal places


Did you know Pythagoras was one of the earliest people known to have given up eating meat for moral reasons?


Now you have an appetite for Pythagoras... let's go!


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