Length of Diagonal of Unit Square

Theorem

The length of a diagonal of a square of side length $1$ is $\sqrt 2$) (the square root of $2$).

Proof

Two adjacent sides $AB$, $BC$ and the diagonal $AC$ of square $ABCD$ form a right triangle.

The hypotenuse of triangle $\triangle ABC$ can be found by using Pythagoras's Theorem:

$AC^2 = AB^2 + BC^2$

from which:

$AC^2 = 2$

and so:

$AC = \sqrt 2$

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$
• 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Paradox
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$