Calculate the diagonal of a square from side length, area, perimeter, or circumradius. Also shows side, area, perimeter, and the diagonal-to-side ratio.
The diagonal of a square is the straight line connecting two opposite vertices. Because a square has four right angles, the Pythagorean theorem gives us the elegant relationship d = a√2, where a is the side length. This means the diagonal is always about 41.4% longer than the side.
Knowing the diagonal is essential in many practical contexts: measuring the screen size of a TV or monitor (which is quoted as the diagonal), checking that a rectangular frame is truly square by measuring diagonals, cutting material on the bias, or computing the turning radius of a square object.
This calculator supports multiple input modes — enter any one of side length, area, perimeter, diagonal itself (for reverse verification), or circumradius — and it computes the diagonal along with all related square properties. The result includes a visual comparison of side vs diagonal, a step-by-step derivation, and a reference table of common square diagonals for quick lookup.
The ratio d/a = √2 ≈ 1.41421356 is one of the most famous irrational numbers in mathematics, known since at least the time of the ancient Babylonians.
This calculator is useful whenever you need to move between the different ways a square can be described without re-deriving the relationship each time. If a plan gives you area, a spec sheet lists perimeter, or a drawing shows only the circumradius, you can still recover the diagonal and every other core dimension immediately. That saves time in drafting, fabrication, classroom work, and measurement checks where the diagonal is the value that matters most.
Diagonal = a√2 Side from diagonal: a = d / √2 Area from diagonal: A = d² / 2 Diagonal from area: d = √(2A) Diagonal from perimeter: d = (P / 4)√2 Circumradius = d / 2 = a√2 / 2
Result: Diagonal ≈ 14.1421 cm
By the Pythagorean theorem: d = √(10² + 10²) = √200 = 10√2 ≈ 14.1421 cm. The area is 100 cm² and d²/2 = 200/2 = 100 confirms consistency.
A square diagonal is more than a textbook line segment. It represents the longest straight-line distance that fits inside the square, so it becomes the critical measurement for clearance, corner-to-corner bracing, screen sizing, and fit checks. Because the diagonal is always $\sqrt{2}$ times the side, even a modest change in side length changes the diagonal noticeably. That constant ratio is what makes square problems predictable and easy to verify once you know any one core measurement.
In practice, you are not always handed the side length. A floor tile might be specified by area, a frame by perimeter, or a design sketch by circumradius. This calculator works backward from each of those values to recover the side and then computes the diagonal from the standard square relationships. That is especially helpful for checking construction drawings, validating geometry homework, or reverse-engineering dimensions from partial specifications without doing several algebra steps by hand.
Square diagonals are used to verify whether a layout is truly square, because matching diagonals confirm consistent geometry. They also show up in monitor sizing, cabinet fit calculations, diagonal braces, and material cuts placed corner to corner. When measuring a real object, use the same unit for every value and compare the returned side, area, and perimeter against the physical dimensions you expect. If one of those looks unreasonable, the issue is usually the original measurement rather than the formula.
The diagonal d = a√2, derived from the Pythagorean theorem applied to the right triangle formed by two sides and the diagonal. Use this as a practical reminder before finalizing the result.
Since A = a², we get a = √A, then d = √A × √2 = √(2A). For example, if A = 50, then d = √100 = 10.
It is always √2 ≈ 1.41421356. This is an irrational number that cannot be expressed as a simple fraction.
A square has exactly two diagonals. They are equal in length, bisect each other at right angles, and each equals a√2.
Yes: A = d²/2. This is sometimes more convenient than converting back to the side first.
The circumradius of a square is half the diagonal: R = d/2 = a√2/2. It is the radius of the circle circumscribing the square.