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What Is a Square?
A square is a four-sided shape (a quadrilateral) with four equal sides and four right angles. It's the most symmetrical of all quadrilaterals — a square is simultaneously a rectangle (opposite sides equal, all angles 90°) and a rhombus (all sides equal), making it the shape that satisfies both definitions at once. Tiles, chess squares, computer pixels, paving slabs and picture frames are all commonly square.
Properties of a Square
- All four sides are equal in length.
- All four interior angles are 90°, totalling 360°.
- Both diagonals are equal in length, bisect each other at 90°, and each bisects a pair of opposite angles.
- It has four lines of symmetry: two through opposite corners, two through the midpoints of opposite sides.
- It has rotational symmetry of order 4 — it looks identical after every 90° turn.
- A square is the regular polygon with four sides (a "regular quadrilateral").
Area Formula
A = s². Multiply the side by itself. A square with 6-unit sides has an area of 36 square units. Because area scales with the square of the side, doubling the side quadruples the area — a common source of estimating errors when scaling plans up or down.
Perimeter Formula
P = 4s. Multiply the side by four, since all four sides are identical. The same 6-unit square has a perimeter of 24 units. See the dedicated perimeter calculator for other shapes.
Diagonal Formula
d = s√2. A square's diagonal splits it into two congruent right isosceles triangles, with both legs equal to the side — so this is the Pythagorean theorem applied to a 45-45-90 triangle: d = √(s² + s²) = s√2. For s = 6, d = 6 × 1.41421 ≈ 8.485 units. See the Pythagorean theorem calculator for the underlying theorem.
Finding the Side Length
| You know | Formula | Example |
|---|---|---|
| Area | s = √A | A = 36 → s = 6 |
| Perimeter | s = P / 4 | P = 24 → s = 6 |
| Diagonal | s = d / √2 | d ≈ 8.485 → s = 6 |
Because a square has only one independent dimension, any single measurement — side, area, perimeter or diagonal — is enough to determine every other property. That's what makes the calculator above able to solve from just one entry, unlike a rectangle, which needs two.
Inscribed Circle
The largest circle that fits entirely inside a square touches the midpoint of each side. Its radius is exactly half the side length: r = s/2. For s = 6, the inscribed circle has a radius of 3. This is the circle a square peg would need to be turned into to fit through the same opening with no gap on any side.
Circumscribed Circle
The smallest circle that passes through all four corners has its centre at the square's centre and a radius equal to half the diagonal: R = d/2 = s√2/2 = s/√2. For s = 6, R ≈ 4.243. The circumscribed circle is always larger than the inscribed one — their ratio is always exactly √2, regardless of the square's size.
Square vs Rectangle
| Property | Square | Rectangle |
|---|---|---|
| Sides | All four equal | Opposite sides equal (l ≠ w generally) |
| Angles | Four right angles | Four right angles |
| Area | A = s² | A = l × w |
| Diagonal | d = s√2 | d = √(l² + w²) |
| Relationship | Every square is a rectangle (the special case l = w); not every rectangle is a square | |
See the full rectangle calculator for the general case with two independent dimensions.
Square vs Rhombus
| Property | Square | Rhombus |
|---|---|---|
| Sides | All four equal | All four equal |
| Angles | Always 90° | Opposite angles equal, not necessarily 90° |
| Diagonals | Equal in length, perpendicular | Perpendicular, but not necessarily equal in length |
| Area | A = s² | A = ½d₁d₂ or base × height |
| Relationship | A square is a rhombus with the extra condition that all angles are 90° | |
Square vs Circle
| Property | Square | Circle |
|---|---|---|
| Area | A = s² | A = πr² |
| Perimeter / circumference | P = 4s | C = 2πr |
| Longest internal line | Diagonal, d = s√2 | Diameter, d = 2r |
| Corners | Four, at 90° | None — continuously curved |
| Efficiency | Tiles and stacks with no gaps | Least perimeter for a given area of any shape |
| Typical applications | Tiles, panels, grid layouts, footings | Pipes, wheels, tanks, manholes |
The two shapes trade off against each other: a square tiles a plane with no wasted space, which is why floors and grids use squares, while a circle encloses the most area for a given perimeter of any shape at all, which is why pipes, tanks and wheels are round. A circle inscribed inside a square touches all four sides and covers about 78.5% of the square's area (π/4); a circle circumscribed around a square passes through all four corners and is about 1.57 times the square's area (π/2). See the circle calculator for circle-specific working.
Step-by-Step Worked Examples
From the side — s = 6
From the area — A = 100
From the perimeter — P = 40
From the diagonal — d = 10
Construction — a square footing pad
Flooring — a square study
Construction tolerance — a footing specified as 1.2 m × 1.2 m ± 5 mm
Real-World Applications
| Field | Use |
|---|---|
| Construction | Square footings, slab pads, structural grid setout |
| Flooring | Square tiles, parquet and panel sizing |
| Landscaping | Raised garden beds, paving slabs, checkerboard patios |
| Manufacturing | Square stock, panels and sheet material |
| Design | Grid layouts, logo design, pixel-based graphics |
| Education | Geometry fundamentals, area-scaling demonstrations |
Glossary
- Side (s)
- The length of one edge — identical on all four sides of a square.
- Area (A)
- The space enclosed by the square, in square units.
- Perimeter (P)
- The total distance around the boundary, in linear units.
- Diagonal (d)
- The line connecting two opposite corners, always longer than a side.
- Inscribed circle
- The largest circle that fits entirely inside the square, touching each side's midpoint.
- Circumscribed circle
- The smallest circle passing through all four corners.
- Interior angle
- The angle inside the square at each corner — always 90°.
- Diagonal angle
- The angle a diagonal makes with a side — always 45° in a square.
- Aspect ratio
- The ratio of length to width — always 1:1 for a square, since all sides match.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing area and perimeter units | Area is squared (m²); perimeter is linear (m) |
| Forgetting to multiply the diagonal formula by √2 | d = s√2, not just s |
| Assuming doubling the side doubles the area | Area scales with the square of the side — it quadruples |
| Mixing up inscribed and circumscribed radius | Inscribed r = s/2; circumscribed R = s/√2, always larger |
| Treating every rhombus as a square | A square needs 90° angles too, not just equal sides |
Formula Cheat Sheet
Quick Reference
Area:A = s² | Perimeter: P = 4s | Diagonal: d = s√2Side from area:
s = √A | from perimeter: s = P/4 | from diagonal: s = d/√2Inscribed radius:
r = s/2 | Circumscribed radius: R = s/√2 = d/2Interior angle:
90° | Diagonal angle: 45° | Aspect ratio: 1:1Scaling:
×k on side → ×k² on area
Practice Questions
Beginner (with answers)
- Find the area of a square with 7 cm sides.
- Find the perimeter of a square with 9 m sides.
- A square has an area of 64. Find its side.
- A square has a perimeter of 32. Find its side.
- Find the diagonal of a square with 5 cm sides (2 dp).
Show answers
1) 7²=49 cm² 2) 4×9=36 m 3) √64=8 4) 32/4=8 5) 5√2≈7.07 cm
Advanced (with answers)
- A square has a diagonal of 20. Find its side, area and perimeter.
- Find the inscribed and circumscribed circle radii for a square with 12 cm sides.
- A square footing pad is 1.5 m × 1.5 m, poured 250 mm deep. Find the concrete volume.
- A square room needs new carpet. If the room is 5 m × 5 m, how many m² should be ordered with 8% waste?
- Two squares have sides in ratio 1:2. What is the ratio of their areas?
Show answers
1) s=20/√2≈14.142, A≈200, P≈56.569 2) r=6 cm, R=12/√2≈8.485 cm 3) 1.5×1.5×0.25=0.5625 m³ 4) 25 m² × 1.08 = 27 m² 5) 1:4, since area scales with the square of the side
🔑 Key Takeaways
- A square needs just one value — side, area, perimeter or diagonal — to be fully solved.
- Area A = s²; perimeter P = 4s; diagonal d = s√2.
- The diagonal is the Pythagorean theorem applied to a 45-45-90 triangle.
- Inscribed circle radius r = s/2; circumscribed radius R = s/√2 — always √2 times larger.
- Every square is both a rectangle and a rhombus, but the reverse isn't true.
Frequently Asked Questions
What is a square?
A square is a four-sided shape with all sides equal and all angles 90°. It's both a special rectangle and a special rhombus — the only shape that satisfies both definitions.
How do you calculate the area of a square?
A = s² — multiply the side by itself. A square with 6-unit sides has an area of 36 square units.
How do you calculate the perimeter of a square?
P = 4s — multiply the side by four. A 6-unit square has a perimeter of 24 units.
How do you find the diagonal of a square?
d = s√2 — the Pythagorean theorem applied to a 45-45-90 triangle formed by two sides and the diagonal. A 6-unit square has a diagonal of about 8.485 units.
What is the square formula?
There are three core formulas: area A = s², perimeter P = 4s, and diagonal d = s√2. Any one of side, area, perimeter or diagonal is enough to find the rest.
How do you find the side length of a square?
From area: s = √A. From perimeter: s = P/4. From diagonal: s = d/√2. Only one of these four values is ever needed, since a square has just one independent dimension.
What is the difference between a square and a rectangle?
A square has all four sides equal; a rectangle only requires opposite sides to be equal. Every square is a rectangle (the special case where length equals width), but a rectangle is only a square when all sides match.
Can a rectangle be a square?
Yes — when a rectangle's length and width are equal, it is by definition also a square.
What is the difference between a square and a rhombus?
Both have four equal sides, but a rhombus's angles don't have to be 90°. A square is a rhombus with the extra condition that every angle is a right angle, which is also why a square's diagonals are equal in length while a general rhombus's usually aren't.
Is every square a rhombus?
Yes — since all four sides are equal, a square automatically satisfies the definition of a rhombus, with the added constraint of right angles.
What is the inscribed circle of a square?
The largest circle that fits entirely inside the square, touching the midpoint of each side. Its radius is exactly half the side length: r = s/2.
What is the circumscribed circle of a square?
The smallest circle passing through all four corners, centred on the square. Its radius equals half the diagonal: R = d/2 = s/√2, always √2 times larger than the inscribed circle's radius.
What is the interior angle of a square?
Always exactly 90°, at every corner, totalling 360° across the four corners.
What is the exterior angle of a square?
Also 90° — the exterior and interior angles of a square are supplementary and, since the interior is 90°, the exterior is 90° too. The four exterior angles always sum to 360° for any convex polygon.
What angle does the diagonal make with a side?
Exactly 45°, since a square's diagonal bisects each 90° corner angle into two equal 45° angles.
What is the aspect ratio of a square?
Always 1:1, because all four sides are the same length.
How is a square used in construction?
Square footings, slab pads and structural grids are common, and checking that a layout is square (using the 3-4-5 method or equal diagonals) is a routine step in setting out any rectangular structure too.
How do you calculate flooring for a square room?
Multiply the side by itself for area (A = s²), then add 8–10% for cuts, waste and pattern matching before ordering.
How do you calculate paving for a square patio?
Find the patio's area with A = s², then divide by the area of one paver to estimate how many are needed, rounding up and allowing extra for cuts around the edges.
What units are used for squares?
Any linear unit for the side, perimeter and diagonal (mm, cm, m, ft); area uses the squared version of the same unit (m², ft²).
Can I calculate square dimensions online?
Yes — enter any one of side, area, perimeter or diagonal above and the calculator solves the rest instantly, with the working shown.
How accurate is a square calculator?
This calculator uses standard double-precision arithmetic, but displayed results are rounded and cannot be more accurate than the measurements entered.
Can students use this calculator?
Yes — enter any one known value and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
What are common square mistakes?
Confusing area (squared units) with perimeter (linear units); forgetting the √2 in the diagonal formula; assuming doubling the side doubles the area, when it actually quadruples it; and mixing up the inscribed and circumscribed circle radii.
Why is square calculation important?
Because squares are common in construction, flooring, paving and manufacturing — getting the area and perimeter right directly affects material orders and cost.
Can this calculator find dimensions from just one value?
Yes — a square has only one independent dimension, so entering any single value (side, area, perimeter or diagonal) is enough to determine every other property automatically.
What happens if I enter more than one value?
The calculator solves from the first value by priority (side, then area, then perimeter, then diagonal) and cross-checks any extra values you entered against the result, flagging a warning if they don't match.
What's the ratio between the inscribed and circumscribed circle?
Always exactly √2 (about 1.414), regardless of the square's size — the circumscribed radius is always √2 times the inscribed radius.
Does doubling the side double the area?
No — it quadruples it, because area scales with the square of the side. A side of 4 gives an area of 16; doubling the side to 8 gives an area of 64, four times as much.
What's the difference between metric and imperial for squares?
The formulas are identical — only the units change. Metric uses mm, cm and m (with m² for area); imperial uses inches, feet and yards (with ft² for area). This calculator supports both.
Can I export the results?
Yes — use Copy Results to copy everything to the clipboard, Export CSV to download a spreadsheet-ready file, or Print / Save as PDF for a printable worksheet with space to sign off.
Does this calculator show step-by-step working?
Yes — every solve shows the formula used, the substituted values, and the derived area, perimeter, diagonal and both circle radii.
About This Calculator
Printable Formula Sheet
A one-page reference with every square formula on this page.
Square Formula Sheet
MegaCalcOnline.com · Area, perimeter, diagonal and circle formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Area | A = s² |
| Perimeter | P = 4s |
| Diagonal | d = s√2 |
| Side from area | s = √A |
| Side from perimeter | s = P / 4 |
| Side from diagonal | s = d / √2 |
| Inscribed circle radius | r = s / 2 |
| Circumscribed circle radius | R = s / √2 = d / 2 |
| Interior angle | 90° |
| Diagonal angle | 45° |
References
- Khan Academy — Quadrilaterals and squares
- OpenStax Mathematics — geometry and measurement
- ACARA — Australian Curriculum: Mathematics
- NIST — SI units for length and area
- Encyclopaedia Britannica — Square (geometry)
- Wolfram MathWorld — Square