Pythagoras in 3D
The familiar theorem of Pythagoras applies to right-angled triangles drawn on a plane surface. In this setting, triangles are made of lines in a two-dimensional space. This idea, however, can be extended by considering lines in three-dimensional space.
Worked example
Example 1
Consider a box-shaped room. Its walls are perpendicular to the floor and the walls that meet are set at right-angles to one another. A rectangular prism can be used to visualise this.
We can construct two lines on the 3D diagram, as shown below.
The diagonal across the floor makes a right-angled triangle with the lines formed where the floor meets two of the walls.
The body diagonal from the floor to the ceiling through the room makes a right-angled triangle with the floor diagonal and the line formed by the intersection of two of the walls.
So, we have constructed two right-angled triangles, and the floor diagonal is common to both. If the wall dimensions are known, then Pythagoras' theorem can be used to first find the length of the floor diagonal. Once the floor diagonal is known, its length can be used to calculate the body diagonal, again using Pythagoras' theorem. This is done in the working after the following diagram:
Firstly, because of the right-angled triangle on the floor of the room, Pythagoras' theorem states that $x^2=d^2+w^2$x2=d2+w2.
Next, because of the right-angled triangle cutting diagonally through the room, Pythagoras' theorem states that $y^2=x^2+h^2$y2=x2+h2.
Putting these results together, the body diagonal length can be calculated using the following formula.
$y^2=d^2+w^2+h^2$y2=d2+w2+h2
Worked example
Example 2
How long is the body diagonal of a box with dimensions $12$12 cm, $19$19 cm and $7$7 cm?
A diagram like the one above would have dimensions $d=12$d=12, $w=19$w=19 and $h=7$h=7. So, the diagonal would be given by $y^2=12^2+19^2+7^2$y2=122+192+72. Thus, $y^2=554$y2=554.
The final step in finding the diagonal is to take the square root. So, $y=\sqrt{554}\approx23.5$y=√554≈23.5 cm.
Practice questions
Question 1
A square prism has sides of length $11$11cm, $11$11cm and $15$15cm as shown.
If the diagonal $HF$HF has a length of $z$z cm, calculate the exact length of $z$z, leaving your answer in surd form.
Now, we want to find $y$y, the length of the diagonal $DF$DF.
Calculate $y$y to two decimal places.
Question 2
A triangular divider has been placed into a box, as shown in the diagram.
If the length of the base of the divider $AC$AC is $z$z cm long, calculate the exact value of $z$z.
Use the exact value of $z$z from part (a) to calculate the length of the diagonal $AD$AD correct to two decimal places.
Calculate the area of the triangular divider correct to two decimal places.
Trigonometry in 3D
Just like Pythagoras' theorem, trigonometry can be applied to right-angled triangles drawn in 3D space.
Practice question
Question 3
A square prism has sides of length $3$3cm, $3$3cm and $14$14cm as shown.
If the diagonal $HF$HF has a length of $z$z cm, calculate $z$z to two decimal places.
If the size of $\angle DFH$∠DFH is $\theta$θ°, find theta to two decimal places.