A square is the special rectangle where both sides are equal, so its area is simply the side squared and its perimeter is four times the side. That squaring relationship is the one worth internalising: a side that is 40% longer gives almost exactly double the area, since 1.4 squared is 1.96.
The diagonal is the side multiplied by the square root of two, roughly 1.4142. This is the oldest known irrational number and the reason A-series paper sizes work: each sheet has sides in the ratio root two to one, so halving it produces the same proportions again. The same constant is why doubling the area of a square means multiplying the side by 1.4142, not by 2.
The area, perimeter and diagonal by side table follows the calculator: 10 rows spanning 1 through 30, there so the result can be cross-checked against a value you already trust. The page opens with side length already set to 9 units, so there is a finished result on screen before anything is typed. The result is also read against 4 bands, from small square at the low end to very large square at the high end, each with a note on what that range means in practice. Most visitors arrive here from, or leave for, Trapezoid Area Calculator and Rectangle Area and Perimeter Calculator.
The formula
s- The side length
A- Area
d- Diagonal
√2- Approximately 1.41421356
How to use this calculator
- Enter the side length of the square.
- The gauge shows the area.
- Perimeter and diagonal appear below.
- The last figure is the side you would need for double the area.
Worked examples
A side of 9
Area is 9 squared = 81, perimeter is 36 and the diagonal is 9 × 1.41421 = 12.7279. The largest circle that fits inside has an area of 63.6173.
Doubling the area
To go from an area of 81 to 162 the side must rise to the square root of 162, which is 12.7279 — the original 9 multiplied by root two, not by two.
Checking the 9 row
The reference table lists Side = 9, giving Area: 81, Perimeter: 36, Diagonal: 12.7279, Inscribed circle area: 63.617. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
Why is the diagonal side times root two?
The diagonal splits the square into two right triangles with legs of length s. Pythagoras gives d squared equals 2s squared, so d is s times root two.
How do I find the side from the area?
Take the square root. An area of 200 square units means a side of 14.142 units.
What proportion of a square does its inscribed circle fill?
Pi over four, or 78.54%, exactly as with any circle in its enclosing square.
Why does 40% longer nearly double the area?
Because 1.4 squared is 1.96, within 2% of two. It is a useful mental shortcut whenever you scale a two-dimensional quantity.
What do the bands under the area mean?
They label the figure so it can be read without a reference point of your own: Small square from 0 to 25; Medium square from 25 to 100; Large square from 100 to 400; Very large square from 400 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Square Area Calculator use?
It computes A = s^2; d = s√2, where s is the side length; A is area; d is diagonal; √2 is approximately 1.41421356. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Square dimensions by side length
| Side | Area | Perimeter | Diagonal | Inscribed circle area |
|---|---|---|---|---|
| 1 | 1 | 4 | 1.4142 | 0.785 |
| 2 | 4 | 8 | 2.8284 | 3.142 |
| 3 | 9 | 12 | 4.2426 | 7.069 |
| 5 | 25 | 20 | 7.0711 | 19.635 |
| 7 | 49 | 28 | 9.8995 | 38.485 |
| 9 | 81 | 36 | 12.7279 | 63.617 |
| 12 | 144 | 48 | 16.9706 | 113.097 |
| 15 | 225 | 60 | 21.2132 | 176.715 |
| 20 | 400 | 80 | 28.2843 | 314.159 |
| 30 | 900 | 120 | 42.4264 | 706.858 |
Area rises with the square of the side while perimeter rises linearly.