Formula and Method for the Diagonal of a Square
A square's diagonal is the line segment connecting two opposite corners. It splits the square into two congruent right triangles, each with legs equal to the side length s and a hypotenuse equal to the diagonal d. By the Pythagorean theorem, d² = s² + s² = 2s², so d = s√2, where √2 ≈ 1.4142. This calculator works in both directions: enter a side length to get the diagonal, or enter a known diagonal to recover the side length, area, and perimeter.
How the calculation works
If you know the side length, multiply it by √2 to get the diagonal: d = s√2. If you know the diagonal instead, divide it by √2 (equivalently, multiply by √2/2 ≈ 0.7071) to recover the side length: s = d/√2. Once the side length is known, the calculator squares it to get the area (A = s², which is also equal to d²/2) and multiplies it by 4 to get the perimeter (P = 4s).
Common mistakes
- Treating the diagonal as a side: the diagonal is always longer than the side — about 41% longer — so using it directly as s in A = s² overstates the area.
- Forgetting to divide by √2: when working backward from a diagonal, dividing by 2 instead of √2 gives a side length that's too small.
- Mixing units: keep the side or diagonal length in one consistent unit throughout — convert inches to feet, or centimeters to meters, before entering the value.
Real-world applications
- Carpentry and framing use the diagonal to "square up" a rectangle or panel — equal diagonals confirm all four corners are true 90° angles.
- Tile and paver layout uses the diagonal to plan diamond (45°) patterns, where the diagonal becomes the effective spacing dimension.
- Screen and panel sizing sometimes specifies a diagonal measurement, from which the true side length and area can be derived.
- Structural bracing calculations use the diagonal length to size a brace that spans a square or rectangular frame.