How this instrument works
A square's diagonal is the straight line between two opposite corners, and it obeys one fixed ratio no matter how large the square is: d = s√2. The derivation needs nothing beyond the Pythagorean theorem applied to the right triangle that a single diagonal always carves out of a square — two adjacent sides play the equal legs, the diagonal plays the hypotenuse, and s² + s² = d² simplifies to d = s√2 because both legs share the same length s.
Draw both diagonals of a square and a second property shows up that a generic rectangle does not share: the two lines cross at a right angle, and each one bisects a 90° corner into a pair of 45° angles. That extra symmetry is the geometric reason the formula collapses to one variable in the first place — the 45° angle sits fixed at every corner of every square, locking the side-to-diagonal ratio at exactly √2 regardless of size.
The relationship scales in the gentlest possible way: double a square's side and the diagonal exactly doubles too, since d = s√2 is linear in s. That is a different story from the square's area, which quadruples under the same doubling — worth keeping straight, since a lot or a screen that looks only a bit bigger corner to corner can enclose far more surface than it first appears. At the vanishing end, a side of 0 correctly collapses the diagonal to 0 as well, with no special case required.
- Enter the square's side length into the Side length field — any unit works, since the formula is a pure ratio of length to length.
- Read the result in Diagonal, the straight-line distance from one corner to the opposite corner.
- To check a different size, overwrite Side length; Diagonal updates immediately with no separate button to press.
- If you only know the diagonal, divide it by √2 (≈1.41421356) to recover the side — the formula box below shows that rearranged step.
Worked example — a side of 5
A square pop-up canopy frame needs a diagonal brace cut to fit corner to corner, and each side of the frame measures exactly 5 units. Multiply by √2 to get the exact length: Diagonal = 5 × √2 = 7.0710678118654755 — the sheet carries the full double-precision value rather than rounding early, so a fabricator working to three decimals reads 7.071 and loses nothing that matters at that scale.
Running the numbers backward checks the cut: a brace measured at 7.0710678118654755 and divided by √2 returns exactly 5, confirming the frame is a true square rather than a slightly stretched rectangle that would rack the fabric off-center. The same ratio holds at any scale — a baseball infield's square base paths run 90 feet a side, so the direct line a catcher's throw travels from home plate to second base is 90 × √2 ≈ 127.28 feet, the diagonal of a much larger square using the identical formula.
Questions
What is the formula for the diagonal of a square?
d = s√2, where s is the side length and √2 ≈ 1.41421356. It follows straight from the Pythagorean theorem: the diagonal is the hypotenuse of a right triangle formed by two adjacent sides, and since both legs equal s, s² + s² = d² collapses to d = s√2 with no separate rule to memorize.
Why isn't a square's diagonal just twice the side length?
Because the diagonal is the straight-line hypotenuse of a right triangle, not the sum of two sides walked in an L-shape. Walking two sides covers 2s of distance, but cutting straight across covers less — s√2 ≈ 1.414s — since a straight line is always the shortest path between two points. Assuming diagonal = 2 × side overstates the true figure by roughly 41%, the most common slip with this formula.
Are a square's two diagonals perpendicular, unlike a plain rectangle's?
Yes. Both diagonals of a square are equal in length, as in any rectangle, but a square's diagonals also cross at a right angle and each one bisects the 90° corners into two 45° angles — properties a plain rectangle's diagonals do not share unless it happens to be a square. That extra symmetry is a direct consequence of all four sides being equal.
How do I find a square's side length from a known diagonal?
Divide by √2: s = d ÷ √2. A measured diagonal of 10 units gives a side of 10 ÷ 1.41421356 ≈ 7.071 units. Rationalizing the denominator turns this into s = d√2 ⁄ 2, the same number reached by a different route, useful if a textbook or a teacher wants that specific form.
Can a square with a whole-number side ever have a whole-number diagonal?
No, never, unless the side itself is zero. Since √2 is irrational, s√2 can only land on a whole number if s carries a compensating irrational factor, which no whole-number or terminating-decimal side ever does. Every clean-integer square still has a diagonal trailing into an endless, non-repeating decimal — 7.0710678118654755… for a side of 5, and no amount of extra digits ever closes it into a fraction.
Does the diagonal grow at the same rate as the square's area?
No — diagonal scales linearly with the side, while area scales with its square. Double a square's side and the diagonal exactly doubles too (5√2 becomes 10√2), but the area quadruples. A side growing tenfold sends the diagonal up tenfold and the area up a hundredfold, the same divergence that makes larger squares feel disproportionately roomier than their outline suggests.