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Square Diagonal Calculator

The diagonal of a square connects opposite corners and is always longer than its sides. Knowing this relationship is essential for geometry, carpentry, construction, and design work. Enter either the side length or diagonal length, and our calculator instantly computes the other using the mathematical relationship between them.

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Creators Mateusz Mucha, BEng
7,584 people find this calculator helpful

Understanding the Diagonal of a Square

A diagonal is a straight line segment connecting two non-adjacent corners of a polygon. Every square has two diagonals, both of equal length, that bisect each other at right angles in the centre. These diagonals are also angle bisectors—each one splits the square's 90° corners into two 45° angles.

When you draw a diagonal across a square, it divides the shape into two identical right triangles. Each triangle has two sides equal to the square's side length and a hypotenuse equal to the diagonal. This geometric relationship is the foundation for the diagonal formula.

The Diagonal Formula

To find the diagonal length, multiply the side length by √2. This relationship comes directly from the Pythagorean theorem, applied to the right triangle formed by two adjacent sides and the diagonal.

diagonal = side × √2

side = diagonal ÷ √2

  • side — The length of one edge of the square
  • diagonal — The distance between two opposite corners

How to Use the Calculator

Enter the known measurement in the corresponding field—either the side length or the diagonal. The calculator immediately displays the unknown value. You can switch between metric and imperial units by clicking the unit selector.

Example: If a square has sides of 10 cm, the diagonal equals 10 × √2 = 14.14 cm. Conversely, if the diagonal is 20 metres, each side is 20 ÷ √2 = 14.14 metres.

The calculator works both directions, making it practical for situations where you know one measurement but need to determine the other.

Practical Applications

Diagonal calculations appear frequently in real-world scenarios. Architects use them to position structural elements and verify that rectangular frames are truly square. Carpenters measure diagonals to ensure cabinet doors and furniture frames are cut correctly. Installation professionals need diagonals when fitting large panels through doorways or around corners.

For screen sizes, monitor diagonals are the standard specification. For flooring and tiling projects, knowing diagonals helps plan layouts and calculate material quantities. Even in landscaping, garden beds and patio areas are often measured this way.

Key Considerations

Keep these points in mind when working with square diagonals.

  1. The √2 factor is constant — No matter the square's size, the diagonal is always 1.414 times the side length. This ratio never changes, which is why the formula is so reliable and universal across all square dimensions.
  2. Measurement precision matters — Small errors in measuring the side accumulate when multiplying by √2. Use precise measurement tools, especially in construction. Rounding too early can compound inaccuracies in larger projects.
  3. Check your work with the Pythagorean theorem — As a sanity check, you can verify: side² + side² = diagonal². For a 5-unit square: 5² + 5² = 50, and √50 = 7.07, confirming the diagonal formula is correct.

Frequently Asked Questions

Why is the diagonal always √2 times the side?

This relationship comes from the Pythagorean theorem. When you draw a diagonal in a square, it creates a right triangle where both legs equal the side length. The diagonal becomes the hypotenuse. Applying the Pythagorean theorem: hypotenuse² = side² + side² = 2 × side². Taking the square root gives hypotenuse = side × √2. This mathematical proof holds true for all squares, regardless of size.

How do I find the side length if I only know the diagonal?

Divide the diagonal by √2 (or approximately by 1.414). For example, if your diagonal measures 10 metres, the side length is 10 ÷ 1.414 = 7.07 metres. If you need a rough estimate without a calculator, dividing by 1.4 gives a close approximation. For perfect accuracy, use the calculator to avoid manual rounding errors.

What is the diagonal of a unit square?

A square with side length 1 has a diagonal of exactly √2, which equals approximately 1.414. This is the simplest case of the diagonal formula, and it's worth memorising as a reference point. All other square diagonals are simply this value multiplied by the side length.

Can I use this for rectangles?

No, this calculator applies only to squares. Rectangles follow a different formula: diagonal = √(length² + width²). For rectangles, you must add the squares of both dimensions, not multiply one by √2. Always confirm your shape is truly square (all sides equal) before using this tool.

Why do the two diagonals of a square have equal length?

Because a square has all four sides of equal length and all angles of 90°, it has perfect symmetry. Both diagonals span the same overall distance from opposite corners. Any non-square rectangle would have diagonals of different lengths if the sides weren't equal, but a square's symmetry guarantees both diagonals are identical.

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