 # Using Pythagoras' Theorem on 3D Shapes

In this worksheet, students will revise Pythagoras' Theorem and learn how to use it in 3D shapes. 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:   ### QUESTION 1 of 10

Pythagoras (top bloke), due to his love of mathematics and science was treated as a god in his time, a bit like Ed Sheeran is today.

Did you know that Pythagoras was in fact responsible for the musical scales we have today. Imagine if Pythagoras' was headlining Glastonbury,  thousands of people hear him introduce his set as...

'And my next number is ..........'

He had many followers (like Ed) who hung on to his every word, they thought his theories were magical. Lets remind our self of his theory. To find the Hypotenuse of a triangle

1. Square one side 2² = 4²

2. Square the other side 6² = 36²

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

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

• you need the length of BC which is given as 5 cm
• you need the length CE which is not given 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

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

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?

Me neither.

Now you have an appetite for Pythagoras... lets go Round your answer to 1 decimal place all the way through the activity

7.2

10.0

9.8

8.5  Calculate the length of  AF

19.64 cm

15.28 cm

20.85cm

14.42 cm Find the length BH

 10.73 cm 11.42 cm 9.98 cm 12.24 cm Length of BH If BE = 12 cm  and BF = 4 cm

Find the length of EF

 10.73 cm 11.42 cm 9.98 cm 12.24 cm Length of BH Calculate the height of this pyramid

23. 24 cm

20.68 cm

22.42 cm

21.86 cm Find the height of this pyramid. What is the height of this cone? What is the length of CE?

10.62 cm

9.43 cm

8.54 cm

11.20 cm AG is 14 cm

EG is 11 cm

Calculate the length of  AE

 7.45 cm 8.32 cm 8.66 cm 7.54 cm Length of AE
• Question 1 Round your answer to 1 decimal place all the way through the activity

10.0
EDDIE SAYS
Did you spot the fact that you needed length AC first? To find AC 6² + 4² = √52 = 7.2 Now you have that information we can crack on. 7² + 7.2² = √ 100.84 = 10.0 to 1 decimal place.
• Question 2 8.5
EDDIE SAYS
First draw the line in that you want to find. A to C Bingo, straight away you should see a right angled triangle. I just love it when that happens. Now apply Pythagoras' theorem as normal 6 x 6 = 36 6 x 6 = 36 36 + 36 = 72 √ 72 = 8.5
• Question 3 Calculate the length of  AF

14.42 cm
EDDIE SAYS
A right angle first time. Excellent. EF is the same as HG 12 cm AE is the same as BF = 8 cm The information is always given to you, you just have to look for it. 12 x 12 = 144 cm² 8 x 8 = 64 cm² 144 + 64 = 208 cm² √208 = 14.42 cm
• Question 4 Find the length BH

 10.73 cm 11.42 cm 9.98 cm 12.24 cm Length of BH
EDDIE SAYS
Once you had found the right angled triangle, did you realise that you had to apply Pythagoras' theorem twice. Twice the work, twice the satisfaction, yeh....well maybe First of all we needed to find FH. 9² + 3² = 90 cm² √ 90 =9.49 (to 2 d.p) 9.49² 90.06 cm² 5² = 25 cm² 90.06 + 25 = 115.06 cm² √ 115.06 = 10.73 cm (to 2 d.p)
• Question 5 If BE = 12 cm  and BF = 4 cm

Find the length of EF

EDDIE SAYS
Oh no this diagram doesn't have numbers on it. No problem, the information is given to us in writing. A bit inconvenient I know, but it is there. This gives us a chance to write on the diagram. Write in what you know BE = 12 cm and BF = 4 cm From this you should see your right angled triangle. 12² - 4² = 128 cm² √ 128 = 11.31 cm Did you spot the subtraction this time? You were finding the shorter side after all
• Question 6 Calculate the height of this pyramid

20.68 cm
EDDIE SAYS
What is going on here. No cubed shaped stuff. It doesn't matter on the shape really as long as you can find the right angled triangle. 22 cm² - 7.5 cm² = 427.75 cm² √427.75 = 20.68 (to 2 d.p) Oh yeh, a sneaky take away as well.
• Question 7 Find the height of this pyramid.

11.26
EDDIE SAYS
I hope you are getting used to this now. Just draw in what you are asked to find. I don't know how people can spot the triangle without drawing on the question... 13² - 6.5² = 126.75 cm² √ 126.75 = 11.26 cm (to 2 d.p)
• Question 8 What is the height of this cone?

EDDIE SAYS
Really, another shape. Yes the beauty of Pythagoras' Theorem is once you can spot a right angle triangle in any shape you can solve the problem. Draw the line in for the height and bingo, there it is. 8 cm² - 3 cm ² = 55 cm² √ 55 = 7.42 (to 2 d.p)
• Question 9 What is the length of CE?

9.43 cm
EDDIE SAYS
Draw in the diagonal to find the length you are looking for and there she is the right angled triangle we want. 8² + 5² = 89 cm² √ 89 = 9.43 cm
• Question 10 AG is 14 cm

EG is 11 cm

Calculate the length of  AE

 7.45 cm 8.32 cm 8.66 cm 7.54 cm Length of AE
EDDIE SAYS
This is just a case of taking the information from a written format instead of it being on a diagram. 14² - 11² = 75 cm² √ 75 = 8.66 cm 3D Pythagoras..no problem..
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