Square Calculator: Area, Perimeter and Diagonal
Enter any one measurement of a square and instantly find all the others. Choose whether you know the side length, area, perimeter or diagonal, type a value, and the calculator works out the remaining three properties in your chosen unit. The step-by-step panel shows the exact arithmetic so you can verify the result or copy it to homework.
Formula
Worked example
A square with a side of 5 m has area = 5² = 25 m², perimeter = 4 x 5 = 20 m, and diagonal = 5 x sqrt(2) = 7.0711 m. Given only the diagonal 7.0711, the side = 7.0711 / sqrt(2) = 5 m, and the area = 5² = 25 m².
Square formulas: area, perimeter and diagonal
A square is a regular quadrilateral: four equal sides and four right angles. Because all sides are the same length a, every other property follows directly. The area A equals a squared, so doubling the side multiplies the area by four. The perimeter P is 4a, growing linearly. The diagonal d is the hypotenuse of the 45-45-90 right triangle formed by any two adjacent sides, so d = a times the square root of 2, approximately 1.4142a. These three relationships mean that any single measurement uniquely determines the whole figure.
How to use the four solve modes
Select which value you already know, type it in, and the calculator fills in the remaining three. If you know the side, the results are immediate: A = a², P = 4a, d = a*sqrt(2). If you know only the area, the side is recovered as a = sqrt(A), then the rest follow. From the perimeter, a = P/4. From the diagonal, a = d/sqrt(2). The step panel shows the substitution with your actual numbers so you can verify each stage. The unit selector lets you work in millimetres, centimetres, metres, kilometres, inches, feet, yards or miles; all four outputs are expressed in that same unit (area in squared units of that choice).
Why the diagonal is always a*sqrt(2)
Draw a diagonal across a square of side a. It cuts the square into two right-angled triangles, each with legs a and a and hypotenuse d. Pythagoras' theorem gives d² = a² + a² = 2a², so d = a*sqrt(2). This is also the formula for the space diagonal of a unit cube restricted to a face, and it appears whenever a 45-degree angle bisects a right-angle corner. A useful check: the diagonal of a square with side 1 metre is always 1.4142 metres, and the diagonal of a square with side 10 cm is always 14.142 cm.
Practical uses of square geometry
Tiles and flooring: divide room area by tile area to find how many tiles you need. Fencing: perimeter gives the total run of fence required. Screen and panel sizing: televisions and monitors are sometimes measured by diagonal, so this calculator converts that diagonal to the equivalent square side if the screen is square. Land surveying: square plots are common, and knowing two values lets you quickly verify the third. Construction: the 3-4-5 right triangle is often used to check square corners, and the diagonal check (measure both diagonals of a rectangle; if they match, the frame is square) relies directly on the Pythagorean relationship above.
Square properties at common side lengths
| Side (a) | Area (a²) | Perimeter (4a) | Diagonal (a × sqrt(2)) |
|---|---|---|---|
| 1 | 1 | 4 | 1.4142 |
| 2 | 4 | 8 | 2.8284 |
| 3 | 9 | 12 | 4.2426 |
| 4 | 16 | 16 | 5.6569 |
| 5 | 25 | 20 | 7.0711 |
| 6 | 36 | 24 | 8.4853 |
| 7 | 49 | 28 | 9.8995 |
| 8 | 64 | 32 | 11.3137 |
| 9 | 81 | 36 | 12.7279 |
| 10 | 100 | 40 | 14.1421 |
How area, perimeter and diagonal scale with side length (a) in any consistent unit.
Frequently asked questions
How do I find the area of a square if I only know the diagonal?
Use the formula A = d² / 2. Because the diagonal is a*sqrt(2), squaring gives d² = 2a², so a² (the area) = d² / 2. For example, a diagonal of 7.0711 m gives an area of 7.0711² / 2 = 50 / 2 = 25 m². This calculator performs the same step: select diagonal, enter the value, and read off the area directly.
What is the perimeter of a square with area 64?
The side is sqrt(64) = 8, so the perimeter is 4 x 8 = 32 (in whatever unit the area is expressed in). The general formula is P = 4 x sqrt(A).
Can I find the side of a square from its perimeter?
Yes, simply divide by 4: a = P / 4. A perimeter of 20 m means a side of 5 m. This works because all four sides are equal.
Why is the diagonal of a square longer than its side?
The diagonal is the hypotenuse of a right triangle with both legs equal to the side. Pythagoras' theorem gives d = a*sqrt(2), and since sqrt(2) is approximately 1.4142, the diagonal is always about 41.4% longer than the side.
How do I check that a frame or room is square using the diagonal?
Measure both diagonals of the rectangle. If they are equal, the corners are 90 degrees and the shape is a true rectangle. If your frame is supposed to be a square (all sides equal), also verify that the diagonal equals the side multiplied by sqrt(2). A 5 m x 5 m square should have diagonals of exactly 7.0711 m.
Does doubling the side double the area?
No. Doubling the side quadruples the area because area scales with the square of the side. A square of side 2 has area 4, while a square of side 4 (double) has area 16, which is four times 4. This is why tiles and paint coverage calculations need careful attention to units and squaring.