What is the formula for the diagonal of a square?

The diagonal of a square is d = s times the square root of 2, where s is the side length. This comes from Pythagoras's theorem, since a square's diagonal splits it into two right-angled triangles with the two equal sides as the legs and the diagonal as the hypotenuse. For example, a square with a 5 metre side has a diagonal of 5 times 1.41421, which is about 7.07 metres.

How do you find the side length from the diagonal?

Rearrange the diagonal formula to solve for the side: s = d divided by the square root of 2, which is the same as s = d times 0.70711. For example, a square with a diagonal of 10 metres has a side length of 10 divided by 1.41421, which is about 7.07 metres.

Is the diagonal of a square always longer than its side?

Yes. Because the square root of 2 is approximately 1.41421, the diagonal is always about 41.4 percent longer than the side length. This ratio holds for every square, regardless of size, since all squares are similar shapes with the same proportions.

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.

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Method note Standard geometric formula d = s√2, derived from Pythagoras's theorem. Same result every time for a given input.

1. Choose What You Know

Solve for

Side length

units

Diagonal length

units

Units can be mm, cm, m or any length unit. Results use the same unit as your input.

2. Formula Used

Diagonal from sided = s × √2

Side from diagonals = d ÷ √2

AreaA = s²

PerimeterP = 4s

√2 (constant used)1.414214

Your Square's Dimensions

Side Length

5.00

Length of each side

Diagonal
7.0711
Corner to corner

Area

25.00

Square units

Perimeter

20.00

Total edge length

Working

Input valueSide = 5.00

Calculation5.00 × √2 (1.414214)

ResultDiagonal = 7.0711

Diagonal to Side Ratio

Diagonal ÷ side1.414214

Percent longer than side41.42%

Two diagonals cross at90 degrees, bisecting each other

Each half-diagonal triangle45-45-90 right triangle

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

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.