Diagonal of a Square Calculator
Find the diagonal of a square from its side length — or work backward from the diagonal, area, or perimeter. Enter any one known value and get every other measurement instantly, with a live square diagram.
✓ 4 calculation modes ✓ Automatic unit conversion ✓ Works on mobile
Diagonal of a Square Calculator
What Is the Diagonal of a Square?
The line that connects opposite corners, and why it always follows one simple ratio.
The diagonal of a square is the line segment that joins two non-adjacent (opposite) corners, cutting straight across the middle of the shape. Every square has exactly two diagonals, and because of the square's perfect symmetry, both diagonals are always equal in length, always bisect each other at a right angle, and always cut each 90° corner exactly in half into two 45° angles.
Each diagonal splits the square into two identical (congruent) right triangles, where the two shorter sides (legs) are simply the square's own sides, and the diagonal itself is the hypotenuse. That relationship means the diagonal can always be found using the Pythagorean theorem — and because both legs of that triangle are equal, the result simplifies to one clean formula: the diagonal is always the side length multiplied by the square root of 2, approximately 1.41421.
This calculator applies that relationship in whichever direction you need it: give it a side length and it returns the diagonal; give it a diagonal and it returns the side; or start from the area or perimeter if that's the measurement you actually have on hand.
Diagonal of a Square Formulas
Four ways to reach the same diagonal, depending on what you already know.
The core formula, derived directly from the Pythagorean theorem applied to the right triangle formed by two sides and a diagonal.
Rearranging the core formula to work backward from a known diagonal to the square's side length.
Useful when you know the square's area rather than its side — since Area = Side², substituting gives this direct shortcut.
Useful when you know the perimeter instead — dividing by 4 first recovers the side length, then the standard formula applies.
How the Diagonal of a Square Calculator Works
Four steps, whichever measurement you start from.
Choose What You Know
Select whether you're starting from the side length, the diagonal, the area, or the perimeter of your square.
Enter the Value
Type in your known measurement using any of seven supported length units, converted internally to metres for consistency.
Automatic Calculation
The calculator applies the correct Pythagorean-based formula instantly and updates the live square-and-diagonal diagram.
Review Every Measurement
See the side, diagonal, area and perimeter all at once, rounded to your chosen decimal precision.
Complete Guide to the Diagonal of a Square
Where the √2 relationship comes from, and how to use it confidently.
Deriving the Diagonal Formula from the Pythagorean Theorem
A square's diagonal splits it into two right triangles, each with two legs equal to the square's side length, a, and a hypotenuse equal to the diagonal. The Pythagorean theorem states that a² + a² = diagonal², which simplifies to diagonal² = 2a², and taking the square root of both sides gives diagonal = a√2. Because √2 is a fixed constant (approximately 1.41421356), this ratio between a square's side and its diagonal never changes, regardless of the square's size.
Working Backward: Diagonal to Side
If the diagonal is known instead of the side, the same relationship rearranges to side = diagonal ÷ √2. It's often easier to multiply by the reciprocal instead: side ≈ diagonal × 0.70711, which avoids dividing by an irrational number and is simple to compute with a basic calculator.
Starting from Area Instead
Since a square's area is side², substituting side = √area into the core formula gives diagonal = √(2 × area) directly, without needing to calculate the side length as an intermediate step. This shortcut is useful whenever a floor plan, land parcel or design spec gives area first and diagonal second.
Starting from Perimeter Instead
A square's perimeter is four times its side length, so side = perimeter ÷ 4. Substituting that into the core formula gives diagonal = (perimeter ÷ 4) × √2, letting you skip straight from a fence length or border measurement to the diagonal without a separate step.
Why the Diagonal Is Always Longer Than the Side
Because √2 is approximately 1.41421, a square's diagonal is always roughly 41% longer than its side — never less, and always by exactly that same ratio no matter how large or small the square is. This is a useful sanity check: if a calculated "diagonal" comes out shorter than the side, or more than about 1.5 times longer, something in the calculation has gone wrong.
Diagonal, Area and the 45-45-90 Triangle
Each of the two triangles created by a square's diagonal is a classic 45-45-90 right triangle — a right triangle with two equal 45° angles. In such a triangle, the hypotenuse is always leg × √2, which is exactly the diagonal-of-a-square relationship viewed from a slightly different angle. This connection is why the diagonal-of-a-square formula and the 45-45-90 triangle rules always agree with each other.
Step-by-Step Diagonal of a Square Examples
Eight worked calculations across all four starting measurements.
Example 1 — From Side Length
A square has a side of 5 inches.
Example 2 — From Side Length (Larger)
A square has a side of 11 inches.
Example 3 — From Diagonal Back to Side
A square has a diagonal of 10 metres.
Example 4 — Unit Square
A square has a side of exactly 1 unit.
Example 5 — Unit Diagonal
A square has a diagonal of exactly 1 unit.
Example 6 — From Area
A square has an area of 25 m².
Example 7 — From Perimeter
A square has a perimeter of 20 m.
Example 8 — Large-Scale Construction
A square foundation slab has a side of 12 metres.
Builders often check this diagonal on-site to confirm the foundation's corners are perfectly square (a "square-check").
Quick Reference Tables
Common side lengths and their diagonals, plus area and perimeter shortcuts.
| Side | Diagonal |
|---|---|
| 1 | 1.414 |
| 2 | 2.828 |
| 5 | 7.071 |
| 10 | 14.142 |
| 11 | 15.556 |
| 20 | 28.284 |
| Area | Side (√Area) | Diagonal |
|---|---|---|
| 4 | 2 | 2.828 |
| 9 | 3 | 4.243 |
| 25 | 5 | 7.071 |
| 100 | 10 | 14.142 |
| Perimeter | Side (÷4) | Diagonal |
|---|---|---|
| 8 | 2 | 2.828 |
| 20 | 5 | 7.071 |
| 40 | 10 | 14.142 |
| 80 | 20 | 28.284 |
Benefits of Using a Diagonal of a Square Calculator
Why reach for a dedicated tool instead of recalculating √2 by hand.
Instant Results
Get the diagonal, side, area and perimeter all at once, without looking up √2 yourself.
Works Both Directions
Start from a side or a diagonal and get an accurate answer either way.
4 Starting Points
Solve from side, diagonal, area, or perimeter — whichever measurement you actually have on hand.
Automatic Unit Conversion
Mix millimetres, metres, feet, and yards freely — everything converts internally without extra steps.
Handy On-Site
Quickly check whether a laid-out square foundation or panel is truly square by comparing measured and calculated diagonals.
Mobile-Friendly
The calculator and diagram adapt cleanly to phones and tablets for quick on-site or classroom use.
Practical Applications of a Square's Diagonal
Where this simple √2 relationship shows up in real-world work.
Construction & Squaring Foundations
Builders measure both diagonals of a rectangular or square layout — if they match, the corners are confirmed to be true right angles.
Furniture & Cabinet Making
Woodworkers check diagonals when assembling square frames, drawers and cabinet carcasses to catch racking before glue-up.
Tiling & Flooring Layouts
Diagonal tile patterns and layout planning rely on knowing the exact diagonal length relative to each tile's side.
Land Surveying & Site Layout
Surveyors use diagonal checks to confirm that staked-out plots and building pads are properly square before construction begins.
Geometry Education
The diagonal-of-a-square formula is a standard early example of applying the Pythagorean theorem and √2 in maths classrooms.
Screen & Panel Sizing
Manufacturers of square panels, screens and frames use the diagonal to specify size, matching how diagonal is conventionally quoted for displays.
Common Diagonal of a Square Mistakes
Avoid these errors when calculating a square's diagonal by hand.
Rounding √2 too early
Using 1.4 instead of 1.41421 introduces a small but avoidable error, especially over larger side lengths.
Confusing diagonal with perimeter
The diagonal cuts across the middle of the square; the perimeter is the total distance around all four sides — they measure completely different things.
Forgetting to take the square root when starting from area
Area is side², so finding the side from area requires a square root first, not a simple division by 2.
Dividing perimeter by the wrong number
A square's perimeter is 4 × side, so recovering the side means dividing by 4, not by 2 or 3.
Mixing units mid-calculation
Combining a side in feet with a target diagonal in metres without converting first produces a meaningless result.
Assuming the shape is a true square without checking
The diagonal = side × √2 formula only holds for a perfect square — a rectangle with unequal sides needs the more general diagonal formula instead.
Frequently Asked Questions
Everything people ask about the diagonal of a square.
What is the formula for the diagonal of a square?
Diagonal = Side × √2. Multiply the square's side length by the square root of 2 (approximately 1.41421) to get the diagonal.
How do I find the side of a square given the diagonal?
Divide the diagonal by √2 (or multiply by approximately 0.70711) to get the side length.
What is the diagonal of a square with side 1?
The diagonal is √2, approximately 1.41421, since Diagonal = 1 × √2.
What is the side of a square with diagonal 1?
The side is √2⁄2, approximately 0.70711, found from Side = 1 ÷ √2.
How do I find the diagonal from the area?
Use Diagonal = √(2 × Area) — this skips the intermediate step of finding the side length first.
How do I find the diagonal from the perimeter?
Divide the perimeter by 4 to get the side, then multiply by √2: Diagonal = (Perimeter ÷ 4) × √2.
Why is the diagonal always longer than the side?
Because √2 is approximately 1.41421, which is always greater than 1 — so the diagonal is always about 41% longer than the side, for every square regardless of size.
Are both diagonals of a square the same length?
Yes. A square's two diagonals are always equal in length, and they bisect each other at a right angle.
How is the diagonal formula related to the Pythagorean theorem?
A square's diagonal is the hypotenuse of a right triangle formed by two of its sides, so a² + a² = diagonal² simplifies directly to Diagonal = Side × √2.
Does this formula work for rectangles too?
No, not directly — a rectangle's diagonal uses √(length² + width²) since its two sides aren't equal. The √2 shortcut only applies when both sides are the same length, as in a square.
Why do builders check a square's diagonals on site?
If a laid-out rectangle or square has matching diagonal measurements, its corners are confirmed to be true right angles — an easy field check without a protractor.
Can this calculator convert between units?
Yes. Every mode accepts millimetres, centimetres, metres, kilometres, inches, feet and yards, converting internally to metres for the calculation.
Is this calculator accurate?
Yes. It applies the exact Pythagorean-derived relationship between a square's side and diagonal, with adjustable decimal precision.
Is it free to use?
Yes, the Diagonal of a Square Calculator is free, with no signup required.
Does it work on mobile?
Yes. The calculator, mode tabs and diagram are fully responsive and work well on phones and tablets.
Calculate a Square's Diagonal Instantly
Enter a side, diagonal, area, or perimeter and get every measurement of your square in seconds.