What it does
You cannot measure √2 with a ruler marked in centimetres, but you can draw it exactly. This page shows two ways.
- Square root spiral. Draw a right triangle with both short sides 1: its long side is √2. On the end of that long side stand another side of length 1 at right angles; the new long side is √3. Keep going and the long sides, the spokes from the centre, are √2, √3, √4 = 2, √5, … The slider sets how many triangles; the newest spoke is drawn in red, and the steps show Pythagoras with your numbers and how far the spiral has turned.
- √x on the number line. For any x: mark AB = x and BC = 1 on a line, draw a semicircle on AC, and raise a line at right angles at B. It meets the semicircle at D, and BD = √x. An arc from B carries that length onto the number line at E.
It fits the Class 9 Maths chapter on the world of numbers (locating √2, √3 and √x on the number line).
The spiral is Pythagoras over and over, as in pythagoras proof playground; to see √2 digit by digit, zoom into it with the number line zoom; for more ruler-and-compass work, try interactive geometry construction.
How to use it
- Pick Square root spiral or √x on the number line.
- For the spiral, move n (1 to 30 triangles). For the number line, move x or type any value up to 100.
- Read Why it works: each step with the lengths from your picture.
A worked example
With n = 5, the spokes are √2, √3, 2, √5 and the newest, √6 ≈ 2.4495. Triangle 5 has legs √5 and 1, so its long side is √(5 + 1) = √6. Its angle at the centre is tan⁻¹(1/√5) = 24.09°, and the five triangles turn 160.92° in all.
With x = 3.5 on the number line: AC = 4.5, so the semicircle has radius 2.25. O is 1.25 from B, so BD² = 2.25² − 1.25² = 5.0625 − 1.5625 = 3.5, and BD = √3.5 ≈ 1.8708.
Tips
- The spiral first completes a full turn with its 17th triangle: 16 triangles turn 351.15°, 17 turn 364.78°.
- The spokes that are whole numbers come at √4, √9, √16, √25: perfect squares.
- The semicircle method works for decimals too, such as √2.5, and for x less than 1.
Check yourself
Four fresh questions: the long side of a spiral triangle, which triangle has a whole-number side, simplifying a square root, and the radius in the semicircle construction. Nothing is saved.
FAQ
Who was Theodorus? Theodorus of Cyrene, a Greek mathematician of the 5th century BCE. Plato says he showed that √3, √5, … up to √17 are irrational; the spiral is named after him, though the drawing itself is modern.
Why does the semicircle give √x? Triangle ABD is similar to triangle DBC, so AB ÷ BD = BD ÷ BC, which means BD² = AB × BC = x × 1.
Is this from the NCERT book? It covers the same topic with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- The spiral of Theodorus (right triangles with legs √k and 1) and the semicircle construction of √x (the geometric mean, from Euclid's Elements Book VI, Proposition 13). Written for fibub.
- Checked against Python: spoke lengths √(k + 1) and the turn after k triangles (Σ tan⁻¹(1/√k) = 351.1504° after 16 and 364.7834° after 17), and the semicircle height √x for 1,000 random x.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.