Square Formulas – Everything You Need
A square is a regular quadrilateral with four equal sides and four right angles (90° each). It is the most symmetric of all rectangles and is a special case of both a rectangle and a rhombus. Every square is a rectangle, but not every rectangle is a square.
Perimeter P = 4s
Diagonal d = s × √2 ≈ s × 1.41421
Side from Area: s = √A
Side from Perimeter: s = P ÷ 4
Side from Diagonal: s = d ÷ √2
Reference Table – All Square Sizes
| Side (s) | Area (s²) | Perimeter (4s) | Diagonal (s√2) |
|---|---|---|---|
| 1 | 1 | 4 | 1.414 |
| 2 | 4 | 8 | 2.828 |
| 3 | 9 | 12 | 4.243 |
| 5 | 25 | 20 | 7.071 |
| 7 | 49 | 28 | 9.899 |
| 10 | 100 | 40 | 14.142 |
| 12 | 144 | 48 | 16.971 |
| 15 | 225 | 60 | 21.213 |
Worked Examples
Example 1 – Tiling a Square Room
A square room has sides of 6 m. How many 30cm × 30cm tiles are needed?
Tile area = 0.3 × 0.3 = 0.09 m²
Tiles needed = 36 ÷ 0.09 = 400 tiles
Example 2 – Find Side from Area
A square garden has an area of 169 m². What is the length of one side and the perimeter?
Perimeter = 4 × 13 = 52 m
Diagonal = 13 × √2 ≈ 18.38 m
Example 3 – Screen Diagonal
A square display has a diagonal of 28.28 cm. What is its side length and area?
Area = 20² = 400 cm²
Why the Diagonal = s√2
The diagonal divides a square into two right isosceles triangles with legs of length s. Applying the Pythagorean theorem: d = √(s² + s²) = √(2s²) = s√2. This is why √2 appears constantly in geometry — it is the ratio of the diagonal to the side in a unit square.
Real-World Applications
- Construction: Square floor tiles, ceiling panels, wall sections
- Art & design: Canvas sizing, pixel grids, checkerboard patterns
- Land measurement: Square plots, sports courts (basketball, tennis)
- Mathematics: Perfect squares in number theory (1, 4, 9, 16, 25...)
- Technology: Aspect-ratio-locked square images on social media, QR codes