Constructing segments of length $\sqrt{13}$ and $\sqrt{22}$, given a segment of length $1$

Question

Given a line segment of $1$ unit in length construct a line segment that is

• $\sqrt{13}$ in length
• $\sqrt{22}$ in length

Is it best to use the root spiral of theodorus or is there a more efficient method? Thanks in advance.

Answer 1

In order to construct a line segment with length $\sqrt{13}$, I would construct it as the hypotenuse of a right triangle such that the length of the other two sides are $2$ and $3$. After that, I would construct another right triangle whose catheti have lengths $\sqrt{13}$ and $3$. The length of its hypotenuse will then be equal to $\sqrt{22}$.

Answer 2

It is easy to construct $\sqrt a$ for each $a$ that is constructible:

(picture from commons.wikimedia)