Worked examples
The question
Question: find the diagonal of a box 3 by 4 by 12.
box 3 by 4 by 12
The diagonal goes from a bottom corner to the top corner on the other side, right through the box.
1 Floor diagonal
First the floor.
Its diagonal d makes a right triangle with the sides 3 and 4.
Pythagoras: d squared is 3 squared plus 4 squared.
3 times 3 is 9.
4 times 4 is 16.
Together 25.
d2 = 32 + 42 = 9 + 16 = 25
a² + b² = c²
5 times 5 is 25.
So d is 5.
d = √25 = 5
Careful!
We cannot just add the sides.
A diagonal is a short cut: Pythagoras, twice.
Careful: not 3 + 4 + 12 = 19 ✗
2 Diagonal of the box
Now a new right triangle stands up: the floor diagonal d, the height 12, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 12 squared.
d squared is 25.
12 times 12 is 144.
Together 169.
D2 = d2 + 122 = 25 + 144 = 169
13 times 13 is 169.
So D = 13.
D = √169 = 13
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
9 plus 16 plus 144 is 169, the same D squared.
So the answer is right!
3² + 4² + 12² = 9 + 16 + 144 = 169
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.
▶ Watch it solved
The question
Question: find the diagonal of a cube with side 2.
cube, side 2
The diagonal goes from a bottom corner to the top corner on the other side, right through the box.
1 Floor diagonal
First the floor.
Its diagonal d makes a right triangle with the sides 2 and 2.
Pythagoras: d squared is 2 squared plus 2 squared.
2 times 2 is 4.
2 times 2 is 4.
Together 8.
d2 = 22 + 22 = 4 + 4 = 8
a² + b² = c²
8 is not a square number.
We do not need d itself: the next step uses d squared.
keep d² = 8
Careful!
We cannot just add the sides.
A diagonal is a short cut: Pythagoras, twice.
Careful: not 2 + 2 + 2 = 6 ✗
2 Diagonal of the box
Now a new right triangle stands up: the floor diagonal d, the height 2, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 2 squared.
d squared is 8.
2 times 2 is 4.
Together 12.
D2 = d2 + 22 = 8 + 4 = 12
12 is not a square number.
A calculator gives 3.464, and so on.
D = √12 = 3.464…
To the nearest tenth, D is about 3.5.
D ≈ 3.5
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
4 plus 4 plus 4 is 12, the same D squared.
So the answer is right!
2² + 2² + 2² = 4 + 4 + 4 = 12
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.
▶ Watch it solved
The question
Question: find the diagonal of a box 5 m by 6 m by 8 m.
box 5 m by 6 m by 8 m
The diagonal goes from a bottom corner to the top corner on the other side, right through the box.
1 Floor diagonal
First the floor.
Its diagonal d makes a right triangle with the sides 5 and 6.
Pythagoras: d squared is 5 squared plus 6 squared.
5 times 5 is 25.
6 times 6 is 36.
Together 61.
d2 = 52 + 62 = 25 + 36 = 61
a² + b² = c²
61 is not a square number.
We do not need d itself: the next step uses d squared.
keep d² = 61
Careful!
We cannot just add the sides.
A diagonal is a short cut: Pythagoras, twice.
Careful: not 5 + 6 + 8 = 19 ✗
2 Diagonal of the box
Now a new right triangle stands up: the floor diagonal d, the height 8, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 8 squared.
d squared is 61.
8 times 8 is 64.
Together 125.
D2 = d2 + 82 = 61 + 64 = 125
125 is not a square number.
A calculator gives 11.180, and so on.
D = √125 = 11.180…
To the nearest tenth, D is about 11.2 m.
D ≈ 11.2 m
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
25 plus 36 plus 64 is 125, the same D squared.
So the answer is right!
5² + 6² + 8² = 25 + 36 + 64 = 125
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.
▶ Watch it solved
