Pythagoras' theorem in space

A box is a solid like a shoebox or a room.

• It has 6 faces – rectangles. Every two faces opposite each other are equal.
• It has 12 edges – the segments where two faces meet – and 8 vertices.
• Its three dimensions are the length $\displaystyle a$, the width $\displaystyle b$ and the height $\displaystyle c$.

The diagonal of a box ($\displaystyle d$) joins a vertex to the vertex farthest from it, through the inside of the box.
We find it with Pythagoras’ theorem, twice:

• The base’s diagonal ($\displaystyle e$): the base is a rectangle, and its diagonal is the hypotenuse of a right-angled triangle with legs $\displaystyle a$ and $\displaystyle b$. So $\displaystyle {e^2}={a^2+b^2}$.
• The box’s diagonal ($\displaystyle d$): the edge $\displaystyle c$ is perpendicular to the base, so to $\displaystyle e$ too. The box’s diagonal is the hypotenuse of a right-angled triangle with legs $\displaystyle e$ and $\displaystyle c$. So $\displaystyle {d^2}={e^2+c^2}$.

We substitute $\displaystyle e^2$ and get:

A box’s dimensions are 3 cm, 4 cm and 12 cm.
How long is its diagonal?

• The base’s diagonal: $\displaystyle {e^2}={3^2+4^2}={9+16}={25}$, so $\displaystyle {e}={5}$ cm.
• The box’s diagonal: $\displaystyle {d^2}={5^2+12^2}={25+144}={169}$, so $\displaystyle {d}={13}$ cm.

Find the length of the box’s diagonal, in cm.

A crate’s dimensions are 30 cm, 40 cm and 120 cm.
Can a stick 125 cm long be put inside it, so that the lid closes?

In the cone too we used Pythagoras’ theorem in space: the height $\displaystyle h$, the radius $\displaystyle r$ and the slant height $\displaystyle l$ are the sides of a right-angled triangle, and $\displaystyle {l^2}={r^2+h^2}$.

A tent is a cone.
The radius of its floor is 1.5 m.
The length of the cloth from the top of the tent to the ground – the slant height – is 2.5 m.
Can a person 1.90 m tall stand up straight in the middle of the tent?

A square pyramid is a solid whose base is a square, and whose four side faces are triangles meeting at one vertex, the apex.

• The apex is right above the centre of the square, $\displaystyle O$.
• $\displaystyle h$ – the pyramid’s height: the segment from the apex to $\displaystyle O$.
• $\displaystyle k$ – the height of a side face: the segment from the apex to $\displaystyle M$, the midpoint of an edge of the base.
• The segment $\displaystyle OM$ is half the base’s side, $\displaystyle \frac{a}{2}$.

The height $\displaystyle h$ is perpendicular to the base, so the segments $\displaystyle h$, $\displaystyle OM$ and $\displaystyle k$ are the sides of a right-angled triangle, and $\displaystyle k$ is its hypotenuse:

The base’s side of a square pyramid is 12 m, and its height is 8 m.
What is the height of a side face?

• $\displaystyle {OM}={12:2}={6}$ m.
• $\displaystyle {k^2}={8^2+6^2}={64+36}={100}$, so $\displaystyle {k}={10}$ m.

The base’s side of a square pyramid is 10 cm, and the height of a side face is 13 cm.
What is the pyramid’s height, in cm?

A sports hall is a box.
Its floor is a rectangle 12 m by 9 m.
It is 8 m high.
A drone flies in a straight line from a corner of the floor to the corner of the ceiling farthest from it.
How many metres does it fly?