Pythagoras’ theorem in 3D - Higher - Pythagoras' theorem - Intermediate & Higher tier - WJEC - GCSE Maths Numeracy (WJEC) Revision

Pythagoras’ theorem in 3D - Higher

Pythagoras’ theorem can also be used to find lengths in 3D spaces.

A pen pot is made out of a cylindrical cup, with a diameter of 7 cm and a height of 10 cm.

Calculate the minimum length a pencil can be, so that it does not fall inside the pot.

As you can see, this will make a right angle at the top corner.

We are trying to find the hypotenuse so we will need to square the sides and add them together to find the square of the hypotenuse.

P² = 10² + 7²

P² = 100 + 49 = 149

P = \(\sqrt{149}\)

P = 12.20655562

The pencil needs to be a minimum of 12.21 cm (to two decimal places).

When using Pythagoras’ theorem to find lengths in 3D spaces, you may need to use it more than once to find the final answer.

Image caption,
Using Pythagoras’ theorem in a 3D space
A room the shape of a cuboid measures 5 m length, 3 m width and 2.5 m height. Calculate the length of the diagonal BH
Image caption,
Using Pythagoras’ theorem in a 3D space
Draw in the line BH and identify the right-angled triangle it is part of
Image caption,
Using Pythagoras’ theorem in a 3D space
Before we can find BH we will need to find FH
Image caption,
Using Pythagoras’ theorem in a 3D space
FH lies in the right-angled triangle FEH
Image caption,
Using Pythagoras’ theorem in a 3D space
We are finding the hypotenuse so need to add
Image caption,
Using Pythagoras’ theorem in a 3D space
Now we know FH, we can calculate BH, the hypotenuse of triangle BFH
Image caption,
Using Pythagoras’ theorem in a 3D space