Using Pythagoras in 3D
Pythagoras' theorem can also be used to solve 3D problems. You often need to use Pythagoras' theorem more than once in each problem as we create extra right-angled triangles in the 3D shapes.
Example
This cuboid has side lengths of 2cm, 3cm and 6cm.
Work out the length of the diagonal AF.
Answer
First use Pythagoras' theorem in triangle ABC to find length AC.
\(A{C^2} = {6^2} + {2^2}\) (as BC must be 2cm)
\(AC = \sqrt {40}\)
You do not need to find the root as we will need to square it in the following step. Next we use Pythagoras' theorem in triangle ACF to find length AF.
\(A{F^2} = A{C^2} + C{F^2}\)
\(AF^{2}=(\sqrt{40})^{2}+3^{2}\)
\(AF^{2}= 40+9\)
\(AF^{2}=49\)
\(AF = \sqrt {49}\)
\(AF = 7cm\)
Now try the example question below.
Question
Work out the vertical height (VM) in this square-based pyramid and give your answer to one decimal place.
The square base ABCD has side lengths of 5cm.
Lengths VA, VB, VC and VD are each 8cm.
M is the mid point of the square base.
First find the diagonal length of the base AC to 1 decimal place.
\(A{C^2} = A{B^2} + A{D^2}\)
\(A{C^2} = {5^2} + {5^2}\)
\(A{C^2} = 25 + 25 = 50\)
\(AC = \sqrt {50}\)
\(AC = 7.07\)
To find AM, we need \(\frac{1}{2}\) of AC.
\(AM = \frac{1}{2} \times AC\)
\(AM = \frac{1}{2} \times 7.07\)
\(AM=3.54\)
Now that we know two dimensions of the triangle VMA inside our pyramid, we can calculate the length of VM.
\(V{M^2} = V{A^2} - A{M^2}\)
\(V{M^2} = {8^2} - {3.54^2}\)
\(V{M^2} = 64 - 12.53\)
\(V{M^2} = 51.47\)
\(VM = \sqrt {51.47}\)
\(VM = 7.2cm\) (to 1 decimal place)
- Remember to use at least one more decimal place than you are asked for during your calculation.
- When giving your final answer, always round according to what the question is asking for, in this case one decimal place.
More guides on this topic
- Determine the gradient of a straight line
- Circle geometry
- Calculating the volume of a standard solid
- Applying the properties of shapes to determine an angle
- Using similarity
- Working with two-dimensional vectors
- Working with three-dimensional coordinates
- Using vector components
- Calculating the magnitude of a vector
- An Approximate History of Co-ordinates