Diagonal of a Square Calculator
This calculator finds the diagonal of a square from its side length, or works backwards from a known diagonal to find the side length, using the geometric relationship d = s times the square root of 2, derived from Pythagoras's theorem. Start by choosing what you know from the dropdown, either the side length or the diagonal, then enter that measurement in any unit you like, such as millimetres, centimetres or metres; the calculator works with the same unit throughout, so results come back in the unit you entered. Once you enter a value, it instantly returns four results: the side length, the diagonal, the area and the perimeter of your square, along with a working panel that shows the exact calculation used and the diagonal-to-side ratio, which is always about 41.4 percent for any square. This makes it useful for practical jobs such as checking that a deck, foundation or room frame is properly square, since a true square has equal diagonals measured corner to corner, as well as for packaging, screen sizing and general maths problems. Because the formula is a fixed mathematical constant rather than an estimate, the results are exact for the values you enter, with rounding applied only for display. Simply switch between solving for the diagonal or the side depending on which measurement you already have, and the calculator handles the rest.
1. Choose What You Know
Units can be mm, cm, m or any length unit. Results use the same unit as your input.
2. Formula Used
Working
Diagonal to Side Ratio
How to Calculate the Diagonal of a Square
A square's diagonal is the straight line connecting two opposite corners. Because all four angles in a square are right angles, drawing a diagonal splits the square into two identical right-angled triangles. Each triangle has the two sides of the square as its legs and the diagonal as its hypotenuse. Applying Pythagoras's theorem (a squared plus b squared equals c squared) with both legs equal to the side length s gives d squared = s squared + s squared = 2s squared, so d = s times the square root of 2.
Worked Example
Take a square with a side length of 5 units. The diagonal is 5 × 1.414214 = 7.071068, which rounds to 7.07 units. The area is 5² = 25 square units, and the perimeter is 4 × 5 = 20 units.
Working Backwards from the Diagonal
If you already know the diagonal and need the side length, divide by the square root of 2 (or multiply by 0.707107). For example, a square with a diagonal of 10 units has a side length of 10 ÷ 1.414214 = 7.071068 units, which rounds to 7.07 units.
| Side Length | Diagonal | Area | Perimeter |
|---|---|---|---|
| 1 | 1.41 | 1 | 4 |
| 5 | 7.07 | 25 | 20 |
| 10 | 14.14 | 100 | 40 |
| 100 | 141.42 | 10,000 | 400 |
Where This Comes Up
Finding a square's diagonal is common in building and trades work, such as checking that a foundation, deck, or room frame is square by comparing the two diagonal measurements (they should be equal). It also comes up in packaging (fitting a square item into a round tube), screen and display sizing, and general geometry problems in maths and stats.
Related Calculators
- Maths and Stats Calculators
- Area of a Square Calculator
- Square Calculator
- Square Root Calculator
- 45-45-90 Triangle Calculator
Sources: Standard Euclidean geometry and Pythagoras's theorem, as taught in the NZ maths curriculum (nzmaths.co.nz). The diagonal to side ratio (√2 ≈ 1.41421) is a fixed mathematical constant.
This calculator gives exact results for a perfect square based on the values you enter. Rounding is applied for display only; the underlying calculation uses full precision.