We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?

To obtain a line segment of irrational length, we use the Baudhāyana–Pythagoras theorem and a geometric construction with ruler and compass.

For example, to construct a line segment of length $\sqrt{2}$ :

Step 1 : Draw a line segment OA of length 1 unit.

Step 2 : At point A, draw a perpendicular AB of length 1 unit.

Step 3 : Join O to B. By the Pythagoras theorem :

⇒ OB² = OA² + AB²

⇒ OB² = 1² + 1²

⇒ OB² = 2

⇒ OB = $\sqrt{2}$.

Step 4 : With O as centre and OB as radius, draw an arc cutting the number line at point P. Then OP = $\sqrt{2}$.

This way, we can construct line segments of any irrational length of the form $\sqrt{n}$ using the Pythagoras theorem repeatedly.

Hence, we can obtain lines of irrational length by using right-angled triangles and applying the Pythagoras theorem.