Calculate the area, perimeter, diagonal, circumradius, and inradius of a square from side length, diagonal, perimeter, or area. Compare with rectangles.
The square is the most fundamental regular quadrilateral — a four-sided polygon where every side is equal and every interior angle is exactly 90°. It is simultaneously a rectangle, a rhombus, and a regular polygon, inheriting the strongest properties of all three families.
The area of a square with side length s is simply s². Its perimeter is 4s, and its diagonal — the line connecting opposite corners — measures s√2. These three quantities are all you need to fully describe the square, yet they interrelate in elegant ways: given any one measurement, you can derive all the others.
Squares also have two important associated circles. The circumscribed circle (circumradius R = s√2 / 2) passes through all four vertices. The inscribed circle (inradius r = s / 2) is tangent to every side. The ratio R / r = √2 is a constant unique to the square among rectangles.
In everyday life, squares appear as floor tiles, coasters, chessboard cells, picture frames, and digital pixels. In mathematics and physics, the square is the basis for area measurement itself — one square meter is the area of a square with 1 m sides. Understanding its properties is essential for carpentry, tiling, fabric cutting, landscaping, and game design.
This calculator lets you start from any one measurement — side, diagonal, perimeter, or area — and instantly derives all others. It includes a comparison tool to see how a rectangle with the same length but different width compares, plus presets and a reference table of common real-world squares.
This calculator is useful when area is only part of the question. In many practical jobs, a square floor, tile, frame, or pad must be priced by area, bordered by perimeter, checked by diagonal, or compared against a rectangle that uses the same length in one direction. Instead of solving each quantity separately, you can derive the full set of square properties from whichever measurement you already have.
It is especially helpful for estimating materials and comparing design choices. If you know a diagonal or a total area, the calculator quickly translates that into side length and perimeter. The built-in rectangle comparison also shows how changing one dimension breaks the square's balance, which is useful for layout planning, classroom demonstrations, and sanity-checking measurements before purchase or construction.
Area: A = s² Perimeter: P = 4s Diagonal: d = s√2 Circumradius: R = s√2 / 2 Inradius (Apothem): r = s / 2 Side from diagonal: s = d / √2 Side from perimeter: s = P / 4 Side from area: s = √A
Result: Area = 100 cm², perimeter = 40 cm, diagonal ≈ 14.1421 cm, with a 10 by 12 rectangle available for comparison.
When mode = side and input = 10, the calculator starts from a square with side length 10 cm. It computes area = 10² = 100 cm², perimeter = 4 × 10 = 40 cm, diagonal = 10√2 ≈ 14.1421 cm, circumradius ≈ 7.0711 cm, and inradius = 5 cm. Setting compareW = 12 also compares that square against a rectangle measuring 10 cm by 12 cm.
For a square, area is often the headline value, but it is rarely the only one that matters. A flooring project may start with square footage, yet trim depends on perimeter and diagonal measurements help verify that a layout is truly square. Because all of these properties come from one side length, an area-focused calculator is most useful when it also exposes the related dimensions that affect material estimates and fit.
One of the most instructive comparisons in elementary geometry is holding one side fixed and changing the other. The moment the width stops matching the length, the square becomes a rectangle and several elegant constants disappear. The diagonal is no longer side × √2, the inradius and circumradius relationships change, and the perimeter-area tradeoff becomes less balanced. That comparison helps explain why squares are such efficient and visually stable shapes.
When you solve from diagonal, perimeter, or area, convert back to side length first. That keeps the rest of the formulas simple and reduces mistakes. Also watch units carefully: perimeter stays linear, area is squared, and any comparison against a rectangle should use the same measurement system for both dimensions. Using the calculator this way reinforces the structure behind the formulas instead of turning the problem into isolated arithmetic.
Area = s², where s is the side length. For example, a square with side 5 m has area 25 m².
Divide the diagonal by √2: s = d/√2. For a diagonal of 10, the side is 10/1.4142 ≈ 7.071.
The diagonal connects opposite corners and equals s√2, where s is the side. It splits the square into two 45-45-90 right triangles.
Yes. A square is a special rectangle where all four sides are equal. It is also a special rhombus (all sides equal) and a regular polygon.
A circle with the same perimeter encloses about 27% more area. The circle is the shape that maximizes area for a given perimeter.
Find the area (s²), divide by the area of one tile, round up, add 10% for waste. Multiply the total tile count by the price per tile.