Square Calculator
Find the side length, area, perimeter, and diagonal of a square. Enter any one value you know and get the other three instantly.
Quick Examples
What Is a Square?
A square is a quadrilateral with four equal sides and four right angles (90° each). It is the most symmetric four-sided shape: a regular quadrilateral. Because all sides are equal, every measurement of a square — its area, perimeter, and diagonal — can be derived from any single one of them. That is exactly what this square calculator does: give it the side length, area, perimeter, or diagonal, and it computes the other three.
Squares appear everywhere: floor tiles, chessboards, picture frames, garden plots, city blocks, and pixels on a screen. Students meet them in geometry class, builders measure them on site, and designers lay them out in grids. Knowing how the four properties relate saves real time whenever only one measurement is at hand.
Square Formulas
Let s be the side length of the square. The three core formulas are:
Where the Diagonal Formula Comes From
The diagonal splits the square into two right triangles, each with legs of length s. By the Pythagorean theorem, d² = s² + s² = 2s², so d = √(2s²) = s√2. Since √2 ≈ 1.4142, the diagonal is always about 41% longer than the side — a handy mental check.
How to Use the Square Calculator
- Choose which measurement you already know: side length, area, perimeter, or diagonal.
- Enter the value (any positive number, including decimals).
- Click Calculate Square. The tool first recovers the side length, then derives the other three properties.
The calculator solves in both directions, so it works whether you are starting from a side measurement or working backward from an area or diagonal.
Worked Examples
Example 1: Given the side length
A square garden bed has sides of 5 m. Its area is 5² = 25 m², its perimeter is 4 × 5 = 20 m, and its diagonal is 5√2 ≈ 7.07 m — the longest straight line you could walk across it.
Example 2: Given the area
A square tile covers 49 cm². The side is √49 = 7 cm, the perimeter is 4 × 7 = 28 cm, and the diagonal is 7√2 ≈ 9.90 cm.
Example 3: Given the perimeter
A square fence is 36 m around. Each side is 36 ÷ 4 = 9 m, the enclosed area is 9² = 81 m², and the diagonal is 9√2 ≈ 12.73 m.
Example 4: Given the diagonal
A square TV screen measures 10 inches corner to corner. The side is 10 ÷ √2 ≈ 7.07 inches, the area is about 50 in², and the perimeter is about 28.28 inches.
Solving Backward: Every Inversion
Because each property determines the side, you can always work backward:
| If you know | Side length s = |
|---|---|
| Area A | √A |
| Perimeter P | P ÷ 4 |
| Diagonal d | d ÷ √2 |
From the side, everything else follows directly. A useful shortcut: the diagonal can be found straight from the area with d = √(2A). For an area of 50, d = √100 = 10.
Real-Life Applications
- Flooring and tiling: convert a room's square footage into tile side counts, or check how many 30 cm tiles fit a 25 m² floor.
- Construction: verify a foundation is truly square by checking that both diagonals are equal — a direct application of d = s√2.
- Gardening: turn a known perimeter of fencing into planting area.
- Design and screens: relate a screen's diagonal (the advertised size) to its actual width and height.
- Quilting and crafts: cut fabric squares from a known area of material with minimal waste.
Common Mistakes
- Confusing perimeter and area: doubling the side doubles the perimeter but quadruples the area. A 10 m square has 4× the area of a 5 m square, not 2×.
- Forgetting square units: area must be reported in cm², m², ft² — never plain cm.
- Guessing the diagonal: the diagonal is not twice the side; it is s√2 ≈ 1.414s. Doubling the side overestimates it by 41%.
- Mixing up square (shape) with squaring (x²): the area formula uses s², which is why the operation "multiply by itself" inherited the name — but this calculator is about the geometric shape.
Frequently Asked Questions
How do you find the area of a square?
Multiply the side length by itself: A = s². A square with side 5 has area 25 square units.
What is the diagonal of a square?
The line connecting opposite corners, with length d = s√2 ≈ 1.4142s. A side-5 square has a diagonal of about 7.07.
How do you find the side length from the area?
Take the square root: s = √A. An area of 49 gives a side of 7.
What is the perimeter of a square?
Four times the side: P = 4s. A side-6 square has perimeter 24.
Is a square a rectangle?
Yes — a square is a rectangle with all sides equal. Every square is a rectangle, but not every rectangle is a square.
How do you find the diagonal if you know the area?
Use d = √(2A). For an area of 50, d = √100 = 10.