7.0711
units
7.0711
units
The Square Diagonal Calculator computes the diagonal length of a square given its side length using the formula d = s√2. The diagonal of a square is a line segment connecting two opposite vertices, and its calculation provides a beautiful connection between square geometry and the irrational number √2 — one of the most historically significant numbers in mathematics.
Every square has two diagonals that are equal in length, perpendicular to each other, and bisect each other at the center of the square. These diagonals also bisect the vertex angles, creating four 45° angles at each vertex. Each diagonal divides the square into two congruent isosceles right triangles (45-45-90 triangles), which are among the most important triangle types in geometry and trigonometry.
The formula d = s√2 is derived directly from the Pythagorean theorem. When a diagonal is drawn in a square with side s, it creates a right triangle with both legs equal to s. Applying the Pythagorean theorem: d² = s² + s² = 2s², therefore d = s√2. This derivation is one of the simplest and most elegant applications of the Pythagorean theorem.
The number √2 ≈ 1.41421356... is historically momentous. According to mathematical tradition, the Pythagorean school in ancient Greece discovered that √2 could not be expressed as a ratio of two integers — it was the first number proven to be irrational. This discovery, possibly made by Hippasus of Metapontum around 450 BCE, challenged the Pythagorean belief that all numbers could be expressed as ratios of whole numbers and fundamentally changed the course of mathematical thought.
The diagonal of a square has important practical applications. In construction and carpentry, knowing the diagonal is essential for cutting materials at 45° angles, setting up square frames, and verifying that structures are truly square. If you know a square has side length s, its diagonal must be exactly s√2. If the measured diagonal differs from this value, the shape is not a true square.
In computer science and digital imaging, the diagonal of a square pixel determines the effective resolution and the distance between diagonally adjacent pixels. Screen measurements, image scaling, and anti-aliasing algorithms all involve diagonal calculations. The √2 ratio also appears in the ISO 216 paper size standard (A-series paper), where each paper size has a √2:1 aspect ratio, ensuring that folding the paper in half produces a geometrically similar rectangle.
The 45-45-90 triangle formed by the diagonal is one of the two "special right triangles" in trigonometry (the other being the 30-60-90 triangle). In a 45-45-90 triangle, the hypotenuse is always √2 times each leg. This relationship gives the exact trigonometric values: sin(45°) = cos(45°) = 1/√2 = √2/2, and tan(45°) = 1. These values are fundamental to trigonometry and are used extensively in physics, engineering, and computer graphics.
This calculator provides instant diagonal computation from the side length. Whether you need to verify construction squareness, calculate material dimensions for 45° cuts, solve geometry problems, or work with 45-45-90 triangles, this tool delivers precise results based on the elegant formula d = s√2.
The Square Diagonal Calculator uses the formula:
$$d = s\sqrt{2}$$
where:
Derivation from the Pythagorean theorem:
The diagonal creates a right triangle with legs s and s:
$$d^2 = s^2 + s^2 = 2s^2$$
$$d = \sqrt{2s^2} = s\sqrt{2}$$
The inverse formula (finding side from diagonal) is:
$$s = \frac{d}{\sqrt{2}} = \frac{d\sqrt{2}}{2}$$
The result is the straight-line distance between opposite corners of the square. It is always exactly √2 ≈ 1.4142 times the side length. This diagonal is also the diameter of the circumscribed circle (the smallest circle that contains the entire square) and serves as the hypotenuse of the 45-45-90 right triangles formed within the square. The diagonal is the longest straight line that can be drawn entirely within the square.
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Diagonal = 10√2 ≈ 14.1421 units. This is also the diameter of the circle circumscribed around the square.
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Diagonal = 1 × √2 ≈ 1.4142 units. This is the fundamental case that defines √2 geometrically — the diagonal of the unit square.
The classic proof (by contradiction) assumes √2 = p/q in lowest terms. Then 2 = p²/q², so p² = 2q², meaning p² is even, so p must be even (p = 2k). Substituting: (2k)² = 2q², so 4k² = 2q², giving q² = 2k², which means q is also even. But if both p and q are even, the fraction was not in lowest terms — a contradiction. Therefore √2 cannot be rational.
Use the inverse formula: s = d/√2 = d√2/2. For example, if the diagonal is 20 units, the side = 20/√2 = 20 × √2/2 = 10√2 ≈ 14.14 units.
A 45-45-90 triangle is an isosceles right triangle where the two acute angles are each 45°. The sides are in the ratio 1 : 1 : √2. The diagonal of a square creates two such triangles. These triangles are fundamental in trigonometry, giving sin(45°) = cos(45°) = √2/2.
Yes, the two diagonals of a square always intersect at right angles (90°). This is a special property of squares and rhombi. In a general rectangle, the diagonals bisect each other but are not perpendicular. The perpendicularity of the diagonals, combined with equal length, is unique to the square among all quadrilaterals.
Since d = s√2, we can express s = d/√2, and therefore A = s² = d²/2. This means the area of a square is half the square of its diagonal. Equivalently, the diagonal of a square equals √(2A), where A is the area.
The √2 ratio appears in ISO paper sizes (A4, A3, etc., which have a √2:1 aspect ratio), in the diagonal of any square (screens, tiles, rooms), in 45° angle calculations, in music theory (equal temperament tuning), and in electrical engineering (RMS to peak voltage ratio for sine waves is √2). It is one of the most practically important irrational numbers.
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