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3D Pythagoras - Higher

3D Pythagoras - Higher

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Tutor: Bilal

Summary

3D Pythagoras

In a nutshell

So far, you have used Pythagoras' theorem to find lengths of 22​ dimensional shapes. However, Pythagoras' theorem can also applied to find distances in 33​ dimensions.



Using Pythagoras' theorem in 33​ dimensions

It is best to show this with a sketch. Here is a cuboid ABCDEFGHABCDEFGH​:



To find the distance AFAF​ (=d=d), first use Pythagoras' theorem on triangle ABHABH​ to find the distance AHAH​, and then use it again on triangle AFHAFH​ to find the distance AFAF​.

Example 1

In the cuboid shown above, a=3cm,b=4cm,d=13cma=3cm,b=4cm,d=13cm. Find the value of cc.


Find AHAH by focusing on the right-angled triangle ABHABH:

a2+b2=(AH)2a^2+b^2=(AH)^2​​​​

32+42=(AH)23^2+4^2=(AH)^2​​

(AH)2=25(AH)^2=25​​

AH=25=5cmAH=\sqrt{25}=5cm​​


Then, find cc by focusing on the triangle AFHAFH:

(AH)2+c2=d2(AH)^2+c^2=d^2​​

52+c2=1325^2+c^2=13^2​​

​​​25+c2=16925+c^2=169​​

c2=144c^2=144​​

c=144=12cmc=\sqrt{144}=12cm​​​​​​

c=12cm\underline{c=12cm}​​

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Other shapes

This formula can be applied to other 3D shapes, such as cubes and pyramids.


Example 2

A cube ABCDEFGHABCDEFGH​ has side lengths 2cm2cm​. What is the exact length of BEBE​ shown in the diagram?


Maths; Trigonometry; KS4 Year 10; 3D Pythagoras - Higher


The length BEBE can be found by using Pythagoras on the triangle BEGBEG. However, to find the length BGBG, first use Pythagoras on the triangle BAGBAG:

(BA)2+(AG)2=(BG)2(BA)^2+(AG)^2=(BG)^2​​


Since it's a cube, all sides are of length 2cm2cm, so substitute this for the lengths BABA and AGAG​:

22+22=(BG)22^2+2^2=(BG)^2​​

(BG)2=8(BG)^2=8​​

BG=8cmBG=\sqrt{8} cm​​


Now, use Pythagoras on the triangle BEGBEG to find BGBG:

(BG)2+(EG)2=(BE)2(BG)^2+(EG)^2=(BE)^2​​

(8)2+22=(BE)2(\sqrt{8})^2+2^2=(BE)^2​​

8+4=(BE)28+4=(BE)^2​​

(BE)2=12(BE)^2=12​​

​​BE=12=23cmBE=\sqrt{12}=2\sqrt{3}cm​​

BE=23cm\underline{BE=2\sqrt{3}cm}​​


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FAQs - Frequently Asked Questions

How do you know when to use Pythagoras' theorem in 3D?

How do you use Pythagoras' theorem in 3 dimensions?

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