Diagonal of a Square Calculator
Enter any one measurement of a square (side length, diagonal, perimeter, or area) and this calculator instantly finds all the others. Switch between metric and imperial units, choose your solve mode, and follow the step-by-step working to see exactly how the Pythagorean theorem connects every dimension of a square.
Formula
Worked example
A square has a side length of 5 m. Diagonal = 5 * sqrt(2) = 5 * 1.41421 = 7.0711 m. Perimeter = 4 * 5 = 20 m. Area = 5^2 = 25 m^2. Reverse: if only the diagonal (7.0711 m) is known, then side = 7.0711 / sqrt(2) = 5.0000 m.
How to calculate the diagonal of a square
Every square is made up of four equal sides meeting at right angles. Drawing a line from one corner to the opposite corner cuts the square into two congruent right triangles. Each triangle has two legs equal to the side length (a) and a hypotenuse equal to the diagonal (d). Applying the Pythagorean theorem gives a^2 + a^2 = d^2, which simplifies to d^2 = 2a^2 and therefore d = a * sqrt(2). Because sqrt(2) is approximately 1.41421, every diagonal is about 41.4 percent longer than its side. The relationship works in reverse too: if you know the diagonal, the side is a = d / sqrt(2).
Solving from any starting measurement
This calculator accepts any one of the four standard measurements of a square and derives the other three. If you know the side, use d = a * sqrt(2). If you know the diagonal, use a = d / sqrt(2). If you know the perimeter (P = 4a), first find the side with a = P / 4, then multiply by sqrt(2). If you know the area (A = a^2), find the side with a = sqrt(A), then multiply by sqrt(2) for the diagonal. Because these relationships are exact, the results carry as much precision as your starting measurement allows. The unit selector converts between millimetres, centimetres, metres, kilometres, inches, feet, and yards so you can work in whatever unit your project uses.
Practical uses of the square diagonal
The diagonal formula appears in many real-world situations. In construction and carpentry, measuring both diagonals of a rectangular frame and checking that they are equal is the quickest way to confirm that all four corners are exactly 90 degrees - a method called "squaring up". In screen and display sizing, manufacturers quote the corner-to-corner diagonal of a square or rectangular panel as its nominal size. In tiling and flooring, knowing the diagonal helps you plan 45-degree layouts and estimate the cut tiles around the edges. In fencing, the diagonal of a square plot tells you the longest straight-line distance across the land. In geometry and physics, the square diagonal appears in lattice calculations, crystal structures, and any problem that involves regular 2D grids.
The 45-45-90 right triangle inside every square
When you draw one diagonal of a square, you create two identical 45-45-90 triangles. The angles are always exactly 45, 45, and 90 degrees regardless of the size of the square, and the sides of these triangles are always in the ratio 1 : 1 : sqrt(2). This is one of the two "special right triangles" taught in trigonometry (the other being the 30-60-90). Because the two legs are equal, this triangle is also isosceles. The 1:1:sqrt(2) ratio means you never need a calculator for the exact relationship - you just need to know whether you are multiplying or dividing by sqrt(2).
Diagonal and area for common square side lengths
| Side length | Diagonal | Perimeter | Area |
|---|---|---|---|
| 1 m | 1.4142 m | 4.0000 m | 1.0000 m^2 |
| 2 m | 2.8284 m | 8.0000 m | 4.0000 m^2 |
| 3 m | 4.2426 m | 12.0000 m | 9.0000 m^2 |
| 5 m | 7.0711 m | 20.0000 m | 25.0000 m^2 |
| 10 m | 14.1421 m | 40.0000 m | 100.000 m^2 |
| 1 ft | 1.4142 ft | 4.0000 ft | 1.0000 ft^2 |
| 3 ft | 4.2426 ft | 12.0000 ft | 9.0000 ft^2 |
| 4 ft | 5.6569 ft | 16.0000 ft | 16.0000 ft^2 |
| 8 ft | 11.3137 ft | 32.0000 ft | 64.0000 ft^2 |
| 10 ft | 14.1421 ft | 40.0000 ft | 100.000 ft^2 |
All values are exact to four decimal places. Diagonal = side * sqrt(2); area = side squared.
Frequently asked questions
What is the formula for the diagonal of a square?
The diagonal of a square with side length a is d = a * sqrt(2), which is approximately d = a * 1.41421. This comes directly from the Pythagorean theorem: the diagonal is the hypotenuse of a right triangle whose two legs are both equal to a, so d^2 = a^2 + a^2 = 2a^2, giving d = a * sqrt(2).
How do I find the side length from the diagonal?
Divide the diagonal by sqrt(2): a = d / sqrt(2), or equivalently a = d * sqrt(2) / 2. For example, if the diagonal is 10 cm, the side is 10 / 1.41421 = 7.0711 cm. This calculator does this automatically when you select the "Diagonal" mode.
How do I find the diagonal from the area?
Take the square root of the area to get the side length (a = sqrt(A)), then multiply by sqrt(2): d = sqrt(2 * A). For example, a square with an area of 36 m^2 has a side of 6 m and a diagonal of 6 * sqrt(2) = 8.4853 m. You can also go directly: d = sqrt(2 * A).
Is the diagonal of a square always longer than the side?
Yes, always. The diagonal equals the side multiplied by sqrt(2) (approximately 1.41421), so it is always about 41.4 percent longer than the side. There is no size of square for which the diagonal could be shorter than or equal to the side.
How is this calculator useful for squaring up a room or frame?
In construction, a room or frame is square (all right angles) if and only if both diagonals are equal in length. You can use this calculator to find what the diagonal should be for a given room side length. Measure both diagonals of your room; if they match the calculated value (and each other), all four corners are at exactly 90 degrees. If they differ, the structure is racked and needs adjustment.
Can I calculate the diagonal of a rectangle with this tool?
This tool is for squares only, where both sides are equal. For a rectangle with sides a and b, the diagonal is sqrt(a^2 + b^2), which requires a separate calculator. If you enter a rectangle side as the "side" here, the result assumes all four sides are that length.