Square
Calculate area, perimeter, and diagonal of a square.
Square Guide
A square is a special case of a rectangle where all sides are of equal length.It is one of two regular polygons that can be inscribed in a circle (alongside the equilateral triangle). All angles in a square are right angles (90 degrees), and the diagonals are equal and intersect at right angles. A square has 4 axes of symmetry and is centrally symmetric about its center.
The area of a square is calculated by squaring the length of the side.If the side has length a, the area is A = a².For example, a square with a side of 6 units has an area of 6² = 36 square units.This formula is simple and intuitive, making it one of the easiest to remember in geometry.
The perimeter of a square is four times the length of the side.The formula is P = 4a.For a side of 6 units, the perimeter is 4 × 6 = 24 units.The perimeter is particularly useful when designing fences, frames, and other elements requiring uniform borders.
The diagonal of a square is longer than the side and equals a√2.This follows from the Pythagorean theorem applied to the right triangle formed by two sides and the diagonal.For a square with side 6, the diagonal has length 6√2 ≈ 8.49 units.The diagonal equals the side multiplied by the square root of two, which is a constant value for all squares.
Practical applications.The square is used everywhere maximum area is needed with minimum perimeter. In architecture, square columns and pillars are most structurally efficient. In board games and puzzles, square tiles are easier to arrange and calculate than other shapes. In mathematics, the square is the basis for defining many other concepts, like square root and exponentiation.
Results are for informational purposes. Verify results with other sources.
Ten metres square is ten times bigger than ten square metres
Two phrases, one word order apart, describing areas that differ by a factor of ten. Ten metres square means a square ten metres on each side — 100 m². Ten square metres is 10 m², a square barely over three metres a side. Builders, landlords and listings use both, and the smaller one is almost always what is on offer.
How it works
- Finds area, perimeter and diagonal of a square from any one of them.
- Works backwards from an area to the side length, which is the useful direction when planning a space.
- Keeps the distinction between a length squared and an area explicit.
area = side² side = √area perimeter = 4 × side diagonal = side × √2 = side × 1.414214 'n metres square' = n × n = n² m² 'n square metres' = n m²
Worked example
The same number read both ways, and what doubling the side does.
- 10 metres square → 100 m²
- 10 square metres → 10 m², a square of 3.16 m a side
- side 2 m → area 4 m²
- side 4 m → area 16 m²
- side 8 m → area 64 m²
Doubling the side quadruples the area, so the two readings diverge fast: at 20 the gap is 400 m² against 20 m². Any quoted area worth money is worth confirming which phrase was meant.
Reading the result
- The diagonal is always the side times √2, which decides whether furniture fits. A 1.00 m square table needs a 1.414 m opening to pass corner-first, and an 0.80 m doorway admits a square of at most 0.566 m a side carried flat.
- Area per metre of fencing improves as the square grows. A 1 m square gives 0.25 m² per metre of fence, a 5 m square gives 1.25 and a 10 m square gives 2.50 — one large enclosure is far cheaper to fence than several small ones of the same total area.
- Going the other way is where intuition fails. Halving a room's area does not halve its side: a 100 m² room becomes 50 m² at 7.07 m a side, not 5 m, because the side scales with the square root.
- The square is the rectangle that encloses the most area for a given perimeter, which is why it turns up wherever material cost tracks the boundary and value tracks the interior.
Common questions
- Is 'ten metres square' the same as 'ten square metres'?
- No — it is ten times larger. Ten metres square is a square with ten-metre sides, so 100 m². Ten square metres is an area of 10 m², a square of about 3.16 m a side. When it matters, ask for the dimensions rather than the phrase.
- If I double the side, how much bigger is the square?
- Four times, not twice. Area goes with the square of the side, so doubling gives four times the area and tripling gives nine times. The same relationship in reverse is why halving an area only shrinks the side by about 29%.