Area of a Square Calculator
This calculator finds the area, perimeter and diagonal of a square from a single side length, using the exact geometric relationships that link a square's four equal sides to everything you need to know about it. Enter the side length and choose a unit, from centimetres, metres, millimetres, kilometres, inches, feet, yards and miles, or type your own custom label such as tiles or blocks. The calculator instantly returns three results: the area (side squared), the perimeter (four times the side) and the diagonal (side multiplied by the square root of two), each shown in a live worked example, a full breakdown of every step, and a plain-English summary sentence. It also lists the formulas used, plus the reverse formulas for working back to the side length from a known area, perimeter or diagonal. Because a square has four equal sides and four right angles, only one measurement is ever needed, and every result updates instantly as you change the side length or unit. Use it for maths homework, tiling and paving quantities, fencing runs, land measurements or any job where you need to move quickly between area, perimeter and diagonal. Results are mathematically exact for the number you enter, so the only uncertainty comes from how precisely you measured the side length in the first place.
1. Square Dimensions
2. Worked Example
For a square with side length 5 m:
The worked example updates with your inputs.
Full Breakdown
Formulas Used
How to Calculate the Area of a Square
A square is a regular quadrilateral: it has four equal sides and four right angles (each 90 degrees). Because all four sides are the same length, only one measurement is needed to fully describe a square.
The three key measurements of a square all follow from the single side length s:
- Area: A = s² (side squared)
- Perimeter: P = 4s (four times the side)
- Diagonal: d = s√2 (by Pythagoras' theorem)
Area Formula Explained
Area measures the amount of two-dimensional space enclosed by a shape. For a square with side length s, the area equals s multiplied by s, because length times width gives area for any rectangle, and a square is a rectangle where length and width are equal.
The result is expressed in square units. If the side is in metres, the area is in square metres (m²). If the side is in centimetres, the area is in square centimetres (cm²), and so on.
Diagonal Formula Explained
The diagonal of a square connects two opposite corners, cutting the square into two right-angled triangles. Each triangle has two shorter sides equal to the side length s and a hypotenuse equal to the diagonal d. Applying Pythagoras' theorem:
d² = s² + s² = 2s²
Therefore d = s√2. The constant √2 is approximately 1.41421. So a 5 m square has a diagonal of 5 x 1.41421 = 7.071 m.
Worked Example: Side of 5 m
| Measurement | Formula | Calculation | Result |
|---|---|---|---|
| Area | A = s² | 5 x 5 | 25 m² |
| Perimeter | P = 4s | 4 x 5 | 20 m |
| Diagonal | d = s√2 | 5 x 1.41421 | 7.071 m |
Common Side Lengths and Their Areas
| Side length | Area | Perimeter | Diagonal |
|---|---|---|---|
| 1 m | 1 m² | 4 m | 1.414 m |
| 2 m | 4 m² | 8 m | 2.828 m |
| 3 m | 9 m² | 12 m | 4.243 m |
| 5 m | 25 m² | 20 m | 7.071 m |
| 10 m | 100 m² | 40 m | 14.142 m |
| 100 m | 10,000 m² | 400 m | 141.421 m |
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Method: Standard Euclidean geometry. Area = s²; Perimeter = 4s; Diagonal = s√2, derived from Pythagoras' theorem (a² + b² = c² with a = b = s). All results are exact in rational arithmetic for integer inputs; the diagonal involves the irrational constant √2.
Results are mathematically exact for the inputs provided. Rounding in the displayed values is for readability only. When applying these results to physical measurements, allow for the precision of your original measurement.