A square is the only four-sided shape fixed by a single number. Know its side, its diagonal, its perimeter or its area, and the other three follow. This square calculator does that conversion, shows the arithmetic, and adds the radius of the circle that fits inside the square and of the circle that passes through its corners.
How to use the square calculator
- Under I know the, tap Side, Diagonal, Perimeter or Area.
- Type the number in Value. An area is in square units; the other three are lengths.
- Choose the Units (inches by default).
- Read the headline: the area, or the side length if you started from the area. The list below gives the side, diagonal, perimeter, area, inscribed circle radius (half the side) and circumscribed circle radius (half the diagonal).
Square formulas
Working back to the side:
The diagonal splits the square into two right isosceles triangles, so by the Pythagorean theorem d² = s² + s² = 2s², and d = s√2. It also means the area is half the diagonal squared, A = d² ÷ 2, handy when the diagonal is the only thing you can measure. The relationship is old: the Babylonian clay tablet known as YBC 7289, nearly 4,000 years old, shows a square with its diagonals and a value for √2 of about 1.414213, within a millionth of the true figure.
Worked example: squaring up a shed pad
You are staking out a 12 ft × 12 ft shed pad and want the corners truly square before the forms go in.
- Choose Side, enter 12 and set the units to feet.
- Diagonal: 12 × √2 ≈ 16.97 ft. The 0.9706 ft part is 0.9706 × 12 ≈ 11.65 in, so on the tape you are aiming for 16 ft 11 5/8 in.
- Adjust the stakes until both diagonals read 16 ft 11 5/8 in. Four equal sides plus equal diagonals guarantee four 90° corners.
- The pad covers 144 ft² and needs 48 ft of form board around the edge.
The diagonal in layout and bracing
Because d = s × √2, a diagonal is always about 41% longer than the side. A brace running corner to corner across a 36 in square opening must be about 50.91 in long. Roof framers use the same ratio: a hip rafter crosses the corner of the building at 45° in plan, so for every 12 in of common-rafter run the hip covers 12 × √2 ≈ 16.97 in, which is why framing tables use 17 in as the hip’s unit run.
The diagonal also links the square to circles. The circle through the four corners has the diagonal as its diameter. That gives the classic sawmill question its answer: the largest square beam you can cut from a 12 in log has a 12 in diagonal, so choose Diagonal, enter 12, and the side comes out at 8.49 in with an area of 72 in². That is 2/π, or about 64%, of the log’s end.
Quick reference
| Side | Diagonal | Perimeter | Area |
|---|---|---|---|
| 1 | 1.4142 | 4 | 1 |
| 2 | 2.8284 | 8 | 4 |
| 3 | 4.2426 | 12 | 9 |
| 4 | 5.6569 | 16 | 16 |
| 6 | 8.4853 | 24 | 36 |
| 8 | 11.314 | 32 | 64 |
| 10 | 14.142 | 40 | 100 |
| 12 | 16.971 | 48 | 144 |
Doubling the side doubles the diagonal and perimeter but quadruples the area.
Unit pitfalls
Area units scale by the square of the length factor. One foot is 12 in, but 1 ft² is 144 in²; one yard is 3 ft, but 1 yd² is 9 ft². Watch the wording too: “10 feet square” means a square 10 ft on a side (100 ft²), while “10 square feet” is a much smaller area.
For shapes with unequal sides, try the rectangle calculator. A square pushed out of shape is a rhombus, covered by the rhombus calculator, and the cube calculator handles the 3D version.
Frequently asked questions
How do I find the diagonal of a square?
Multiply the side by √2, which is about 1.41421. A 12 ft square has a diagonal of 12 × 1.41421 ≈ 16.97 ft, or 16 ft 11 5/8 in to the nearest sixteenth of an inch.
How do I get the side length from the area?
Take the square root of the area. A 400 ft² square room is √400 = 20 ft on each side, so it has an 80 ft perimeter and a diagonal of about 28.28 ft.
What is the difference between 10 square feet and 10 feet square?
Ten square feet is an area of 10 ft², a patch only about 3.16 ft on each side. Ten feet square describes a square that is 10 ft on each side, which covers 100 ft².
What is the largest square I can cut from a circle?
The square's diagonal equals the circle's diameter, so its side is the diameter divided by √2. A 12 in round log yields a beam about 8.49 in square with a 72 in² cross-section, roughly 64% of the log's end area.
Is a square also a rectangle and a rhombus?
Yes. It has four right angles like a rectangle and four equal sides like a rhombus, so their formulas still apply. The rhombus rule, area = product of the diagonals ÷ 2, becomes A = d² ÷ 2 for a square.