Square Diagonal
Calculate the diagonal length of any square instantly from its side using d = asqrt2. Free online square diagonal calculator for students, teachers, and professionals with instant results.
About This Calculator
The Square Diagonal Calculator computes the diagonal length of any square from its side length using the formula d = a x sqrt2, where a is the side length of the square and sqrt2 (approximately 1.414) is the square root of 2. This relationship comes from the Pythagorean theorem: the diagonal divides a square into two congruent right triangles, with the diagonal serving as the hypotenuse.
How the Formula Works
In a square, all four sides are equal, and all interior angles are 90 degrees. The diagonal connects two opposite vertices, forming the hypotenuse of a right triangle where the two legs are the sides of the square. By the Pythagorean theorem: d^2 = a^2 + a^2 = 2a^2, so d = asqrt2. This simple but powerful relationship makes it easy to find the diagonal of any square.
Who Uses This Calculator
Students use it for geometry homework and exam preparation. Teachers use it to verify answers and create practice problems. Architects, carpenters, and construction professionals use it for layout and measurement tasks. DIY enthusiasts use it for home improvement projects like tiling, framing, and cutting square materials.
Regional Notes
Global: The diagonal formula d = asqrt2 is universal and applies regardless of measurement system -- metric (cm, m) or imperial (inches, feet). Simply enter the side length in any unit; the result is returned in the same unit. This calculator works worldwide for any square measurement scenario.
Frequently Asked Questions
What is the formula for the diagonal of a square?
The diagonal of a square is calculated using the formula d = asqrt2, where a is the side length of the square and sqrt2 is approximately 1.414. This is derived from the Pythagorean theorem since the diagonal divides the square into two right triangles.
How do I find the diagonal of a square if I only know the area?
If you know the area of the square, use the formula d = sqrt(2 x area). Since area = a^2, substituting and solving gives d = sqrt(2 x area). For example, a square with area 25 square units has a diagonal of sqrt(2 x 25) = sqrt50 ≈ 7.07 units.
How do I find the side of a square if I only know the diagonal?
To find the side length from the diagonal, divide the diagonal by sqrt2: a = d / sqrt2. For example, if the diagonal is 10 units, the side length is 10 / 1.4142 ≈ 7.07 units.
What is the diagonal of a square with side length 1?
The diagonal of a square with side length 1 is sqrt2, which equals approximately 1.414 units. This is the simplest case of the diagonal formula d = asqrt2 where a = 1.
What is the diagonal of a square with side length 5?
The diagonal of a square with side length 5 is 5sqrt2, which equals approximately 7.071 units. You can verify this by entering 5 into the Square Diagonal Calculator above.
Are both diagonals of a square equal in length?
Yes, both diagonals of a square are always equal in length. Each diagonal measures asqrt2 where a is the side length. The diagonals also bisect each other at right angles and bisect the square's interior angles (45 deg each).
Is the Square Diagonal Calculator free to use?
Yes, the Square Diagonal Calculator on Calculy is completely free to use with no registration or limits. You can calculate as many square diagonals as you need for homework, projects, or professional work.