Square formulas (and what they mean)

A square is the most “symmetrical” rectangle: all four sides are equal and all four angles are 90°. Because everything repeats, you don’t need lots of inputs — one measurement is enough to rebuild the entire shape. That’s why square problems show up everywhere in math classes and in the real world (tile, screens, frames, grids, and more).

• Side → Perimeter: P = 4s
• Side → Area: A = s²
• Side → Diagonal: d = s√2
• Diagonal → Side: s = d / √2

The first two are intuitive: perimeter is the distance around the square, so it’s just four equal sides. area is the size of the surface inside, so it’s “side × side,” which is why you square the unit (cm becomes cm², feet becomes ft², etc.).

The diagonal formula is where students often pause, so here’s the clean explanation: Draw the diagonal and you’ve split the square into two identical right triangles. Each triangle has legs of length s and s, and the diagonal is the hypotenuse. By the Pythagorean theorem, d² = s² + s² = 2s², so d = s√2. This is also why you’ll see the famous constant √2 ≈ 1.41421356… in many square/diagonal problems.

• Perimeter → Side: s = P / 4
• Area → Side: s = √A
• Area → Perimeter: P = 4√A
• Area → Diagonal: d = √(2A) (because d = √2·√A)
• Perimeter → Area: A = (P/4)²
• Perimeter → Diagonal: d = (P/4)√2

Notice how everything is connected. If you can get the side length from the one value you know, the rest is easy. That’s exactly what this calculator does behind the scenes: Step 1: Convert your input into side length s. Step 2: Compute A, P, and d from s.

Also, watch the units: If side is in meters, diagonal and perimeter are also in meters, but area is in square meters (m²). The calculator prints the correct “unit vs unit²” labeling automatically.

Worked examples you can copy

Suppose a square tile has side length s = 12 cm.
Perimeter: P = 4s = 4×12 = 48 cm
Area: A = s² = 12² = 144 cm²
Diagonal: d = s√2 = 12×1.414… ≈ 16.97 cm

A picture frame is a perfect square and the perimeter is P = 80 in.
Side: s = P/4 = 80/4 = 20 in
Area: A = s² = 20² = 400 in²
Diagonal: d = s√2 = 20×1.414… ≈ 28.28 in

A square garden bed covers A = 81 ft².
Side: s = √A = √81 = 9 ft
Perimeter: P = 4s = 36 ft
Diagonal: d = s√2 ≈ 9×1.414… ≈ 12.73 ft

A square screen measures d = 55 cm corner-to-corner.
Side: s = d/√2 = 55/1.414… ≈ 38.89 cm
Perimeter: P = 4s ≈ 155.56 cm
Area: A = s² ≈ 1512.35 cm²

Tiny pro tip: when a diagonal looks “messy,” it’s often because √2 is irrational. That’s normal — it means the exact answer is something like 12√2, and the decimal is just an approximation.

What this calculator does (step-by-step)

The calculator uses one simple idea: everything flows through the side length. No matter what you input (area, perimeter, diagonal, or side), the script first computes the side s. Then it computes perimeter, area, and diagonal from s.

You choose whether your number represents side, area, perimeter, or diagonal. This matters because the same number means totally different things. For example, “36” could be 36 inches of perimeter (which would be a 9-inch side), or it could be 36 square inches of area (which would be a 6-inch side).

• If you know side: s = value
• If you know perimeter: s = value/4
• If you know area: s = √value
• If you know diagonal: s = value/√2

• P = 4s
• A = s²
• d = s√2

Your output is shown in a clean, screenshot-friendly block. You can copy the results as text, use the native share button on mobile, or share directly to WhatsApp/Telegram/Twitter/X/Facebook/LinkedIn. If you hit “Save Result,” it stores your last few square calculations in your browser (local storage) so you can compare sizes.

That’s it. It’s fast because it doesn’t call a server — it’s just math in your browser.

Frequently Asked Questions

• What is the formula for the area of a square?

  The area is A = s², where s is the side length. If the side is 8 cm, the area is 64 cm².

• How do I find the diagonal of a square?

  Use d = s√2. This comes from the Pythagorean theorem because the diagonal forms a right triangle with legs s and s.

• If I know area, how do I get side length?

  Take the square root: s = √A. Example: if A = 49, then s = 7.

• Why is √2 showing up in my result?

  √2 appears because a square’s diagonal is the hypotenuse of a right triangle with equal legs. Since √2 is irrational, the diagonal often has a “never-ending” decimal — that’s normal.

• Does the unit matter?

  Yes. Side, perimeter, and diagonal use normal units (cm, inches, feet). Area uses squared units (cm², in², ft²). Always keep measurements in the same unit system before calculating.

• Can a square have a diagonal longer than twice the side?

  No. Since d = s√2 and √2 ≈ 1.414, the diagonal is always about 1.414× the side — never 2×.

• What’s the fastest way to estimate diagonal without a calculator?

  Multiply side by 1.4 for a quick rough estimate. Example: side 50 → diagonal ≈ 70. Exact would be about 70.71.